how hot can a heat bath get?

Digital Object Identifier (DOI) 101007s00220-009-0857-6Commun Math Phys 292 131ndash177 (2009) Communications in

MathematicalPhysics

How Hot Can a Heat Bath Get

Martin Hairer12

1 Mathematics Institute The University of Warwick Coventry CV4 7ALUnited Kingdom E-mail MHairerWarwickacuk

2 Courant Institute New York University New York NY 10012 USAE-mail mhairercimsnyuedu

Received 2 December 2008 Accepted 31 March 2009Published online 25 June 2009 ndash copy Springer-Verlag 2009

Abstract We study a model of two interacting Hamiltonian particles subject to acommon potential in contact with two Langevin heat reservoirs one at finite and oneat infinite temperature This is a toy model for lsquoextremersquo non-equilibrium statisticalmechanics We provide a full picture of the long-time behaviour of such a systemincluding the existencenon-existence of a non-equilibrium steady state the precise tailbehaviour of the energy in such a state as well as the speed of convergence toward thesteady state

Despite its apparent simplicity this model exhibits a surprisingly rich variety of longtime behaviours depending on the parameter regime if the surrounding potential is lsquotoostiffrsquo then no stationary state can exist In the softer regimes the tails of the energyin the stationary state can be either algebraic fractional exponential or exponentialCorrespondingly the speed of convergence to the stationary state can be either alge-braic stretched exponential or exponential Regarding both types of claims we obtainmatching upper and lower bounds

Contents

1 Introduction 1312 Heuristic Derivation of the Main Results 1383 A Potpourri of Test Function Techniques 1414 Existence and Non-existence of an Invariant Probability Measure 1465 Integrability Properties of the Invariant Measure 1566 Convergence Speed Towards the Invariant Measure 1677 The Case of a Weak Pinning Potential 170

1 Introduction

The aim of this work is to provide a detailed investigation of the dynamic and thelong-time behaviour of the following model Consider two point particles moving in a

132 M Hairer

potential V1 and interacting through a harmonic force that is the Hamiltonian systemwith Hamiltonian

H(p q) = p20 + p2

1

2+ V1(q0) + V1(q1) + V2(q0 minus q1) V2(q) = α

2q2 (11)

We assume that the first particle is in contact with a Langevin heat path at temperatureT gt 0 The second particle is also assumed to have a stochastic force acting on it butno corresponding friction term so that it is at lsquoinfinite temperaturersquo The correspondingequations of motion are

dqi = pi dt i = 0 1dp0 = minusV prime1(q0) dt + α(q1 minus q0) dt minus γ p0 dt +

radic2γ T dw0(t) (12)

dp1 = minusV prime1(q1) dt + α(q0 minus q1) dt +radic

2γ Tinfin dw1(t)

where w0 and w1 are two independent Wiener processes Although we use the symbolTinfin in the diffusion coefficient appearing in the second oscillator this should not beinterpreted as a physical temperature since the corresponding friction term is missingso that detailed balance does not hold even if Tinfin = T We also assume without furthermention throughout this article that the parameters α γ T and Tinfin appearing in themodel (12) are all strictly positive

The equations of motion (12) determine a diffusion on R4 with generator L givenby

L = X H minus γ p0partp0 + γ(

T0part2p0

+ Tinfinpart2p1

) (13)

where X H is the Liouville operator associated to H ie the first-order differential oper-ator corresponding to the Hamiltonian vector field It is easy to show that (12) has aunique global solution for every initial condition since the evolution of the total energyis controlled by

LH = γ (T0 + Tinfin)minus γ p20 (14)

and so EH(t) le H(0) exp (γ (T0 + Tinfin)t) Schematically the system under consider-ation can thus be depicted as follows where we show the three terms contributing to thechange of the total energy

This model is very closely related to the toy model of heat conduction previously stud-ied by various authors in [EPR99bEPR99aEH00RT00RT02EH03Car07HM08a]consisting of a chain of N anharmonic oscillators coupled at its endpoints to two heat

How Hot Can a Heat Bath Get 133

baths at possibly different temperatures The main difference is that the present modeldoes not have any friction term on the second particle This is similar in spirit to thesystem considered in [MTVE02DMP+07] where the authors study the stationary stateof a lsquoresonant duorsquo with forcing on one degree of freedom and dissipation on anotherone Because of this lack of dissipation even the existence of a stationary state is notobvious at all in such a system Indeed if the coupling constant α is equal to zero onecan easily check that the invariant measure for (12) is (formally) given by exp(minus(p2

02+V1(q0))T ) dp0 dp1 dq0 dq1 which is obviously not integrable

One of the main questions of interest for such a system is therefore to understandthe mechanism of energy dissipation In this sense this is a prime example of a lsquohyp-ocoerciversquo system where the dissipation mechanism does not act on all the degreesof freedom of the system directly but is transmitted to them indirectly through thedynamic [Vil07Vil08] This is somewhat analogous to lsquohypoellipticrsquo systems where itis the smoothing mechanism that is transmitted to all degrees of freedom through thedynamic The system under consideration happens to be hypoelliptic as well but this isnot going to cause any particular difficulty and will not be the main focus of the presentwork

Furthermore since one of the heat baths is at lsquoinfinitersquo temperature even if a stationarystate exists one would not necessarily expect it to behave even roughly like exp(minusβH)for some effective inverse temperature β It is therefore of independent interest to studythe tail behaviour of the energy of (12) in its stationary state

In order to simplify our analysis we are going to limit our investigation to one of thesimplest possible cases where V1 is a perturbation of a homogeneous potential Moreprecisely we assume that V1 is an even function of class C2 such that

V1(x) = |x |2k

2k+ R1(x)

with a remainder term R1 such that

supxisinR[minus11]

supmle2

|R(m)1 (x)||x |2kminus1minusm

ltinfin

Here k isin R is a parameter describing the lsquostiffnessrsquo of the individual oscillators (In thecase k = 0 we assume that V1(x) = C + R1(x) for some constant C )

In the case where both ends of the chain are at finite temperature (which would cor-respond to the situation depicted above) it was shown in [EPR99bEH00RT02Car07]that provided that the coupling potential V2 grows at least as fast at infinity as the pinningpotential V1 and that the latter grows at least linearly (ie provided that 1

2 le k le 1 withour notations) the Markov semigroup associated to the model has a unique invariant

134 M Hairer

measure micro and its transition probabilities converge to micro at exponential speed Onecan actually show even more namely that the Markov semigroup consists of compactoperators in some suitably weighted space of functions

Intuitively the condition that V2 grows at least as fast as V1 can be understood bythe fact that in this case at high energies the interaction dominates so that no energycan get lsquotrappedrsquo in the system Therefore the system is sufficiently stiff so that if theenergy of any one of its oscillators is large then the energy of all of the oscillators mustbe large after a very short time As a consequence the system behaves like a lsquomoleculersquoat some effective temperature that moves in the global potential V1 While the argumentspresented in [RT02Car07] do not cover the case of one of the heat baths being at infinitetemperature it is nevertheless possible to show that in this case the Markov semigroupPt generated by solutions to (12) behave qualitatively like in the case of finite tempera-ture In particular if V1 grows at least linearly at infinity the system possesses a spectralgap in a space of functions weighted by a weight function lsquoclose torsquo exp(β0 H) for someβ0 gt 0

This discussion suggests that

1 If V2 V1 1 our toy model can sustain arbitrarily large energy currents2 In this case even though the heat bath to the right is at infinite temperature the

system stabilises at some finite lsquoeffective temperaturersquo as expressed by the fact thatH has finite exponential moments under the invariant measure

This is in stark contrast with the behaviour encountered when V1 grows faster thanV2 at infinity In this case the interaction between neighbouring particles is suppressedat high energies which precisely favours the trapping of energy in the bulk of the chainIt was shown in [HM08a] that this can lead in many cases to a loss of compactness ofthe semigroup generated by the dynamic and the appearance of essential spectrum at 1This is a manifestation of the fact that energy transport is very weak in such systemsdue to the appearance of lsquobreathersrsquo localised structures that only decay very slowly[MA94] In this case one expects that the long-time behaviour of (12) depends muchmore strongly on the fine details of the model For example regarding the finiteness ofthe lsquotemperaturersquo of the second oscillator one may introduce the following notions byincreasing order of strength

1 There exists an invariant probability measure micro for (12) that is a positive solutionto Llowastmicro = 0

2 There exists an invariant probability measuremicro and the average energy of the secondoscillator is finite under micro

3 There exists an invariant probability measure micro and the energy of the second oscil-lator has some finite exponential moment under micro

We will show that it is possible to find parameters such that the second oscillator does nothave finite temperature according to any of these notions of finiteness On the other handit is also possible to find parameters such that it does have finite temperature accordingto some notions and not to others

It turns out that maybe rather surprisingly for such a simple model there are five dif-ferent critical values for the strength k of the pinning potential V1 that separate betweenqualitatively different behaviours regarding both the integrability properties of the invari-ant measuremicro and the speed of convergence of transition probabilities towards it Thesecritical values are k = 0 k = 1

2 k = 1 k = 43 and k = 2 More precisely there exists


a constant C gt 0 such that setting

ζ = 3

4

α2C minus TinfinTinfin

κ = 2

kminus 1 (15)

the results in this article can be summarised as follows

Theorem 11 The integrability properties of the invariant measure micro for (12) and thespeed of convergence of transition probabilities of (12) toward micro can be described bythe following table

Parameter range Integrability of micro Convergence speed Prefactork gt 2 mdash mdash mdashk = 2 Tinfin gt α2C mdash mdash mdashk = 2 Tinfin lt α2C Hζplusmnε tminusζplusmnε Hζ+ε+1

43 le k lt 2 exp(γplusmnHκ ) exp

(minusγplusmntκ(1minusκ)

)exp(δHκ )

1 lt k le 43 exp(γplusmnHκ ) exp(minusγplusmnt) exp(δH1minusκ )

k = 1 exp(γplusmnH) exp(minusγplusmnt) Hε

12 le k lt 1 exp(γplusmnH) exp(minusγplusmnt) exp(δH

1kminus1

)

0 lt k le 12 exp(γplusmnH) exp

(minusγplusmntk(1minusk)

)exp(δH)

k le 0 mdash mdash mdash

Here the symbol lsquomdashrsquo means that no invariant probability measure exists for the cor-responding range of parameters Whenever there exists a (necessarily unique) invariantmeasure micro we indicate integrability functions Iplusmn(H) convergence speeds ψplusmn(t) anda prefactor K (H) The constant ε can be made arbitrarily small whereas the constantsγ+ γminus and δ are fixed and depend on the fine details of the model For each line in thistable the following statements hold

bull One hasint

R4 I+(H(x)) micro(dx) = +infin butint

R4 Iminus(H(x)) micro(dx) lt +infinbull There exists a constant C such that

Pt (x middot )minus microTV le C K (H(x))ψ+(t) (16)

for every initial condition x isin R4 and every time t ge 0bull For every initial condition x isin R4 there exists a constant Cx and a sequence of times

tn increasing to infinity such that

Ptn (x middot )minus microTV ge Cxψminus(tn)for every n

Remark 12 This table is valid only in the case of a lsquochainrsquo consisting of two oscillatorsHowever combining the heuristics of Sect 2 with the formal calculation from [HM08aSect 2] we can conjecture that in the general case of a chain of n + 1 oscillators oneobtains a similar table with κ = 2n

k + 1 minus 2n In particular one would then expect tohave non-existence of an invariant probability measure as soon as k gt 2n

2nminus1

Remark 13 The case k = 2 and Tinfin = α2C is not covered by these results We expectthat the system admits no invariant probability measure in this case In the regimek asymp 2 the heavy tails of the invariant measure as well as the slow relaxation speedsuggest the appearance of intermittent behaviour This can be verified numerically (seeFig 1) and dovetails nicely with the intermittent behaviour that was already observed in[MTVE02DMP+07]

136 M Hairer

0 02 04 06 08 1 12

0

50

100 Energy

Time (times105)

0 02 04 06 08 1 12

0

1000

2000

3000Energy

Time (times105)

0 02 04 06 08 1 12

0

05

1

Energy (in millions)

Time ( times105)

Fig 1 Numerical simulation of the time evolution of the total energy in the case k = 11 (top) k = 18(middle) and k = 25 (bottom) One clearly sees the appearance of intermittency at k asymp 2 followed by thelack of a stationary solution for k gt 2 The numerics was performed with a Stoumlrmer-Verlet scheme that wasmodified to take into account the damping and the stochastic forcing

Remark 14 For k isin (0 12 ) even the gradient dynamic fails to exhibit a spectral gap

It is therefore not surprising (see for example [HN05]) that in this case we see againsubexponential relaxation speeds

Remark 15 This table exhibits a symmetry κ harr k and Hκ harr H around k = 1 (indi-cated by a grayed row in the table) The reason for this symmetry will be explainedin Sect 2 below If we had chosen V1(x) = K log x + R1(x) in the case k = 0 thissymmetry would have extended to this case via the correspondence ζ harr K

T +Tinfin minus 12

Remark 16 It follows from (16) that the time it takes for the transition probabilities

starting from x to satisfy Pt (x middot ) minus microTV le 12 say is bounded by H(x)2minus 2

k for

k isin (1 2) and by H(x)1kminus1 for k isin (0 1) These bounds are expected to be sharp in

view of the heuristics given in Sect 2 below

Remark 17 Instead of considering only distances in total variation between probabilitymeasures we could also have obtained bounds in weighted norms similarly to [DFG06]


Remark 18 The operator (13) appears to be very closely related to the kineticFokker-Planck operator

LV2 = p partq minusnablaV (q) partp minus γ p partp + part2p

for the potential V (q0 q1) = V1(q0) + V1(q1) + α2 (q0 minus q1)

2 The fundamental differ-ence however is that there is a lack of friction on the second degree of freedom Theeffect of this is dramatic since the results from [HN04] (see also [DV01]) show that onehas exponential return to equilibrium for the kinetic Fokker-Planck operator in the casek ge 1 which is clearly not the case here

Finally the techniques presented in this article also shed some light on the mecha-nisms at play in the Helffer-Nier conjecture [HN05 Conj 12] namely that the long-timebehaviour of the Fokker-Planck operator without inertia

LV1 = minusnablaV (q) partq + part2q

is qualitatively the same as that of the kinetic Fokker-Planck operator If V grows fasterthan quadratically at infinity (so that in particular LV1 has a spectral gap) then thedeterministic motion on the energy levels gets increasingly fast at high energies so thatthe angular variables get washed out and the heuristics from Sect 21 below suggeststhat the total energy of the system behaves like the square of an Ornstein-Uhlenbeckprocess thus leading to a spectral gap for LV2 as well

If on the other hand V grows slower than quadratically at infinity then the motion ofthe momentum variable happens on a faster timescale at high energies than that of theposition variable The heuristics from Sect 22 below then suggests that the dynamiccorresponding to LV2 is indeed very well approximated at high energies by that corre-sponding to LV1

These considerations suggest that any counterexample to the Helffer-Nier conjecturewould come from a potential that has very irregular (oscillating) behaviour at infinityso that none of these two arguments quite works On the other hand any proof of theconjecture would have to carefully glue together both arguments

The structure of the remainder of this article is the following First in Sect 2 wederive in a heuristic way reduced equations for the energies of the two oscillators Whilethis section is very far from rigorous it allows to understand the results presented aboveby linking the behaviour of (12) to that of the diffusion

d X = minusηXσ dt +radic

2 dW (t) X ge 1

for suitable constants η and σ The remainder of the article is devoted to the proof of Theorem 11 which is broken

into five sections In Sect 3 we introduce the technical tools that are used to obtainthe above statements These tools are technically quite straightforward and are all basedon the existence of test functions with certain properties The whole art is to constructsuitable test functions in a relatively systematic manner This is done by refining thetechniques developed in [HM08a] and based on ideas from homogenisation theory

In Sect 4 we proceed to showing that k = 2 and Tinfin = α2C is the borderline casefor the existence of an invariant measure In Sect 5 we then show sharp integrabilityproperties of the invariant measure for the regime k gt 1 when it exists This will implyin particular that even though the effective temperature of the first oscillator is alwaysfinite (for whatever measure of finiteness) the one of the second oscillator need not

138 M Hairer

necessarily be In particular note that it follows from Theorem 11 that the borderlinecase for the integrability of the energy of the second oscillator in the invariant measureis given by k = 2 and Tinfin = 7

3α2C These two sections form the lsquomeatrsquo of the paper

In Sect 6 we make use of the integrability results obtained previously in order toobtain bounds both from above and from below on the convergence of transition proba-bilities towards the invariant measure The upper bounds are based on a recent criterionfrom [DFG06BCG08] while the lower bounds are based on a simple criterion thatexploits the knowledge that certain functions of the energy fail to be integrable in theinvariant measure Finally in Sect 7 we obtain the results for the case k le 1 Whilethese final results are based on the same techniques as the remainder of the articlethe construction of the relevant test functions in this case in inspired by the argumentspresented in [RT02Car07]

11 Notations In the remainder of this article we will use the symbol C to denotea generic strictly positive constant that unless stated explicitly depends only on thedetails of the model (12) and can change from line to line even within the same blockof equations

2 Heuristic Derivation of the Main Results

In this section we give a heuristic derivation of the results of Theorem 11 Since weare interested in the tail behaviour of the energy in the stationary state an importantingredient of the analysis is to isolate the lsquoworst-casersquo degree of freedom of (12) thatwould be some degree of freedom X which dominates the behaviour of the energy atinfinity The aim of this section is to argue that it is always possible to find such a degreeof freedom (but what X really describes depends on the details of the model and inparticular on the value of k) and that for large values of X it satisfies asymptotically anequation of the type


2 dW (t) (21)

for some exponent σ and some constant η gt 0 Before we proceed with this pro-gramme let us consider the model (21) on the set X ge 1 with reflected boundaryconditions at X = 1 The invariant measure micro for (21) then has density proportional toexp(minusηXσ+1(σ + 1)) for σ gt minus1 and to Xminusη for σ = minus1 In particular (21) admitsan invariant probability measure if and only if σ gt minus1 or σ = minus1 and η gt 1 For such amodel we have the following result which is a slight refinement of the results obtainedin [Ver00VK04Ver06]

Theorem 21 The long-time behaviour of (21) is described by the following table

Parameter range Integrability of micro Convergence speed Prefactorσ lt minus1 mdash mdash mdashσ = minus1 η le 1 mdash mdash mdash

σ = minus1 η gt 1 Xηminus1plusmnε t1minusη

2 plusmnε Xη+1+ε

minus1 lt σ lt 0 exp(γplusmnXσ+1

)exp

(minusγplusmnt(1+σ)(1minusσ)) exp

(δXσ+1

)

0 le σ lt 1 exp(γplusmnXσ+1

)exp(minusγplusmnt) exp

(δX1minusσ )

σ = 1 exp(γplusmnXσ+1

)exp(minusγplusmnt) Xε

σ gt 1 exp(γplusmnXσ+1

)exp(minusγplusmnt) 1


The entries of this table have the same meaning as in Theorem 11 with the exceptionthat the lower bounds on the convergence speed toward the invariant measure hold forall t gt 0 rather than only for a subsequence

Proof The case 0 le σ le 1 is very well-known (one can simply apply Theorem 34below with either V (X) = exp(δX1minusσ ) for δ small enough in the case σ lt 1 or withXε in the case σ = 1) The case σ gt 1 follows from the fact that in this case one canfind a constant C gt 0 such that EX (1) le C independently of the initial condition

The bounds for σ = minus1 and η gt 1 can be found in [Ver00FR05Ver06] (a slightlyweaker upper bound can also be found in [RW01]) However as shown in [BCG08]the upper bound can also be retrieved by using Theorem 35 below with a test functionbehaving like Xη+1+ε for an arbitrarily small value of ε The lower bound on the otherhand can be obtained from Theorem 36 by using a test function behaving like Xα butwith α 1 (These bounds could actually be slightly improved by choosing test func-tions of the form Xη+1(log X)β for the upper bound and exp((log X)β) for the lowerbound)

The upper bound for the case σ isin (minus1 0) can be found in [VK04] and more recentlyin [DFG06BCG08] This and the corresponding lower bound can be obtained simi-larly to above from Theorems 35 and 36 by considering test functions of the formexp(aXσ+1) for suitable values of a (small for the upper bound and large for the lowerbound) 13

Returning to the problem of interest it was already noted in [EH00RT02] that k = 1is a boundary between two types of completely different behaviours for the dynamic(12) The remainder of this section is therefore divided into two subsections where weanalyse the behaviour of these two regimes

21 The case k gt 1 When k gt 1 the pinning potential V1 is stronger than the cou-pling potential V2 Therefore in this regime one would expect the dynamic of the twooscillators to approximately decouple at very high energies [HM08a] This suggeststhat one should be able to find functions H0 and H1 describing the energies of the twooscillators such that H0 is distributed approximately according to exp(minusH0T ) whilethe distribution of H1 has heavier tails since that oscillator is not directly damped

In order to guess the behaviour of H1 at high energies note first that since H0 isexpected to have exponential tails the regime of interest is that where H1 is very largewhile H0 is of order one In this regime the second oscillator feels mainly its pinningpotential so that its motion is well approximated by the motion of a single free oscil-lator moving in the potential |q|2k2k A simple calculation shows that such a motion

is periodic with frequency proportional to H12minus 1

2k1 and with amplitude proportional to

H1

2k1 In other words one can find periodic functions P and Q such that in the regime

of interest one has (up to phases)

q1(t) asymp H1

2k1 Q(H

12minus 1

2k1 t) p1(t) asymp H

12

1 P(H12minus 1

2k1 t) (22)

Let now 13p and 13q be the unique solutions to 13q = 13p 13p = Q that average out tozero over one period It is apparent from the equations of motion (12) that if we assume

140 M Hairer

that (22) is a good model for the dynamic of the second oscillator then the motion ofthe first oscillator can at least to lowest order be described by

p0(t) = p0(t)minus αH1kminus 1

21 13p(H

12minus 1

2k1 t) q0(t) = q0(t)minus αH

32kminus1

1 13q(H12minus 1

2k1 t)

(23)

where the functions p0 and q0 do not show any highly oscillatory behaviour anymoreFurthermore they then satisfy at least to lowest order the decoupled Langevin equation

dq0 asymp p0 dt d p0 asymp minusV prime1(q0) dt minus αq0 dt minus γ p0 dt +radic

2γ T dw0(t) (24)

that indeed has exp(minusH0T ) as invariant measure provided that we set

H0 = p20

2+ V1(q0) +

α

2q2

0

Let us now return to the question of the behaviour of energy dissipation The averagerate of change of the total energy of our system is described by (14) Plugging our ansatz(23) into this equation and using the fact that 13p is highly oscillatory and averages outto 0 we obtain

LH asymp γ (T + Tinfin)minus γ p20 minus γα2 H

2kminus1

1 〈132p〉

On the other hand it follows from (24) that one has

LH0 asymp γ T minus γ p20 asymp C1 minus C2 H0 (25)

so that one expects to obtain for the energy of the second oscillator the expression

LH1 asymp γ Tinfin minus γα2〈132p〉H

2kminus1

1

This suggests that at least in the regime of interest and since the p-dependence of H1

probably goes likep2

12 the energy of the second oscillator follows a decoupled equation

of the type

d H1 asymp(γ Tinfin minus γα2〈132

p〉H2kminus1

1

)dt +

radic2γ TinfinK H1 dw1(t) (26)

where K is the average of p21 over one period of the free dynamic at energy 1 which

will be shown in (510) below to be given by K = 2k(1 + k)In order to analyse (26) it is convenient to introduce the variable X given by

X2 = 4H1(γ TinfinK ) so that its evolution is given by

d X =(

2

Kminus 1

)1

Xminusradicγα2〈132

p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)

This shows that there is a transition at k = 2 For k gt 2 we recover (21) with σ = minus1and η = 1 minus 2

K lt 1 so that one does not expect to have an invariant measure thusrecovering the corresponding statement in Theorem 11


At k = 2 we still have σ = minus1 but we obtain

η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2

so that one expects to have existence of an invariant probability measure if and onlyif Tinfin lt α2〈132

p〉 Furthermore we recover from Theorem 21 the integrability resultsand convergence rates of Theorem 11 noting that one has the formal correspondenceζ = (η minus 1)2 This correspondence comes from the fact that H asymp X2 in the regimeof interest and that Xηminus1 is the borderline for non-integrability with respect to micro inTheorem 21

In the regime k isin (1 2) the first term in the right-hand side of (27) is negligibleso that we have the case σ = 4

k minus 3 Applying Theorem 21 then immediately allowsto derive the corresponding integrability and convergence results from Theorem 11noting that one has the formal correspondence κ = (σ + 1)2

22 The case k lt 1 This case is much more straightforward to analyse When k lt 1the coupling potential V2 is stiffer than the pinning potential V1 Therefore one expectsthe two particles to behave like a single particle moving in the potential V1 This suggeststhat the lsquoworst casersquo degree of freedom should be the centre of mass of the system thusmotivating the change of coordinates

Q = q0 + q1

2 q = q1 minus q0

2

Fixing Q and writing y = (q p0 p1) for the remaining coordinates we see that thereexist matrices A and B and a vector v such that y approximately satisfies the equation

dy asymp Ay dt + V prime1(Q)v dt + B dw(t)

Here we made the approximation V prime1(q0) asymp V prime1(q1) asymp V prime1(Q) which is expected to bejustified in the regime of interest (Q large and y of order one) This shows that for Qfixed the law of y is approximately Gaussian with covariance of order one and meanproportional to V prime1(Q) Since d Q = (p0 + p1)2 dt we thus expect that over sufficientlylong time intervals the dynamic of Q is well approximated by

d Q asymp minusC1V prime1(Q) dt + C2 dW (t) asymp minusC1 Q|Q|2kminus2 dt + C2 dW (t)

for some positive constants C1 C2 and some Wiener process W We are therefore reducedagain to the case of Theorem 21 with X prop |Q| and σ = 2k minus 1 Since in the regimeconsidered here one has H asymp X2k this immediately allows to recover the results ofTheorem 11 for the case k lt 1

3 A Potpourri of Test Function Techniques

In this section we present the abstract results on which all the integrability and non-integrability results in this article are based as well as the techniques allowing to obtainupper and lower bounds on convergence rates toward the invariant measure All of theseresults without exception are based on the existence of test functions with certain prop-erties In this sense we follow to its bitter end the Lyapunov function-based approachadvocated in [BCG08CGWW07CGGR08] and use it to derive not only upper boundson convergence rates but also lower bounds

142 M Hairer

While most of these results from this section are known in the literature (except forthe one giving the lower bounds on the convergence of transition probabilities whichappears to be new despite its relative triviality) the main interest of the present article isto provide tools for the construction of suitable test functions in problems where differ-ent timescales are present at the regimes relevant for the tail behaviour of the invariantmeasure

The general framework of this section is that of a Stratonovich diffusion on Rn withsmooth coefficients

dx(t) = f0(x) dt +msum

i=1

fi (x) dwi (t) x(0) = x0 isin Rn (31)

Here we assume that f j Rn rarr Rn are Cinfin vector fields on Rn and thewi are indepen-dent standard Wiener processes Denote by L the generator of (31) that is the differentialoperator given by

L = X0 +1

2

msum

i=1

X2i X j = f j (x)nablax

We make the following two standing assumptions which can easily be verified in thecontext of the model presented in the introduction

Assumption 1 There exists a smooth function H Rn rarr R+ with compact level setsand a constant C gt 0 such that the bound LH le C(1 + H) holds

This assumption ensures that (31) has a unique global strong solution We further-more assume that

Assumption 2 Houmlrmanderrsquos lsquobracket conditionrsquo holds at every point in Rn In otherwords consider the families Ak (with k ge 0) of vector fields defined recursively byA0 = f1 fm and

Ak+1 = Ak cup [ f j g] g isin Ak j = 0 mDefine furthermore the subspaces Ainfin(x) = spang(x) existk gt 0 with g isin Ak Wethen assume that Ainfin(x) = Rn for every x isin Rn

As a consequence of Houmlrmanderrsquos celebrated lsquosums of squaresrsquo theorem [Houmlr67Houmlr85] this assumption ensures that transition probabilities for (31) have smooth den-sities pt (x y)with respect to Lebesgue measure In our case Assumption 2 can be seento hold because the coupling potential is harmonic

Assumption 3 The origin is reachable for the control problem associated to (31) Thatis given any x0 isin Rn and any r gt 0 there exists a time T gt 0 and a smooth controlu isin Cinfin([0 T ]Rm) such that the solution to the ordinary differential equation

dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0

satisfies z(T ) le r


The fact that Assumption 3 also holds in our case is an immediate consequence ofthe results in [EPR99aHai05] Assumptions 2 and 3 taken together imply that

1 The operator L satisfies a strong maximum principle in the following sense LetD sub Rn be a compact domain with smooth boundary such that 0 isin D Let further-more u isin C2(D) be such that Lu(x) le 0 for x in the interior of D and u(x) ge 0 forx isin partD Then one has u(x) ge 0 for all x isin D see [Bon69 Theorem 32]

2 The Markov semigroup associated to (31) admits at most one invariant probabilitymeasure [DPZ96] Furthermore if such an invariant measure exists then it has asmooth density with respect to Lebesgue measure

31 Integrability properties of the invariant measure Throughout the article we aregoing to use the following criterion for the existence of an invariant measure with certainintegrability properties

Theorem 31 Consider the diffusion (31) and let Assumptions 2 and 3 hold If thereexists a C2 function V Rn rarr [1infin) such that lim sup|x |rarrinfin LV (x) lt 0 then thereexists a unique invariant probability measure micro for (31) Furthermore |LV | is inte-grable against micro and

int LV (x)micro(dx) = 0

Proof The proof is a continuous-time version of the results in [MT93 Chap 14] Seealso for example [HM08a] 13

The condition given in Theorem 31 is actually an if and only if condition but theother implication does not appear at first sight to be directly useful However it is pos-sible to combine the strong maximum principle with a Lyapunov-type criterion to ruleout in certain cases the existence of a function V as in Theorem 31 This is the contentof the next theorem which provides a constructive criterion for the non-existence of aninvariant probability measure with certain integrability properties

Theorem 32 Consider the diffusion (31) and let Assumptions 1 2 and 3 hold Letfurthermore F Rn rarr [1infin) be a continuous weight function Assume that there existtwo C2 functions W1 and W2 such that

bull The function W1 grows in some direction that is lim sup|x |rarrinfinW1(x) = infinbull There exists R gt 0 such that W2(x) gt 0 for |x | gt Rbull The function W2 is substantially larger than W1 in the sense that there exists a positive

function H with lim|x |rarrinfin H(x) = +infin and such that

lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0

bull There exists R gt 0 such that LW1(x) ge 0 and LW2(x) le F(x) for |x | gt R

Then the Markov process generated by solutions to (31) does not admit any invariantmeasure micro such that

intF(x) micro(dx) ltinfin

Proof The existence of an invariant measure that integrates F is equivalent to the exis-tence of a positive C2 function V such that LV le minus F outside of some compact set[MT93 Chap 14] The proof of the claim is then a straightforward extension of theproof given for the case F equiv 1 by Wonham in [Won66] 13Remark 33 If one is able to choose F equiv 1 in Theorem 32 then its conclusion is thatthe system under consideration does not admit any invariant probability measure

144 M Hairer

32 Convergence speed toward the invariant measure upper bounds We still assumein this section that we are in the same setting as previously and that Assumptions 2ndash3hold The strongest kind of convergence result that one can hope to obtain is exponentialconvergence toward a unique invariant measure In order to formulate a result of thistype given a positive function V we define a weighted norm on measurable functionsby

ϕV = supxisinRn

|ϕ(x)|1 + V (x)

We denote the corresponding Banach space by Bb(Rn V ) Furthermore given a Markovsemigroup Pt over Rn we say that Pt has a spectral gap in Bb(Rn V ) if there exists aprobability measure micro on Rn and constants C and γ gt 0 such that the bound

Ptϕ minus micro(ϕ)V le Ceminusγ tϕ minus micro(V )V

holds for every ϕ isin Bb(Rn V ) We will also say that a C2 function V Rn rarr R+ is aLyapunov function for (31) if lim|x |rarrinfin V (x) = infin and there exists a strictly positiveconstant c such that

LV le minuscV

holds outside of some compact setWith this notation we have the following version of Harrisrsquo theorem [MT93] (see

also [HM08b] for an elementary proof)

Theorem 34 Consider the diffusion (31) and let Assumptions 2 and 3 hold If thereexists a Lyapunov function V for (31) then Pt admits a spectral gap in Bb(Rn V ) Inparticular (31) admits a unique invariant measuremicro

intV dmicro ltinfin and convergence

of transition probabilities towards micro is exponential with prefactor V

However there are situations where exponential convergence does simply not takeplace In such situations one cannot hope to be able to find a Lyapunov function asabove but it is still possible in general to find a ϕ-Lyapunov function V in the followingsense Given a function ϕ R+ rarr R+ we say that a C2 function V Rn rarr R+ is aϕ-Lyapunov function if the bound

LV le minusϕ(V )holds outside of some compact set and if lim|x |rarrinfin V (x) = infin If such a ϕ-Lyapunovfunction exists upper bounds on convergence rates toward the invariant measure can beobtained by applying the following criterion from [DFG06BCG08] (see also [FR05])

Theorem 35 Consider the diffusion (31) and let Assumptions 2 and 3 hold Assumethat there exists a ϕ-Lyapunov function V for (31) where ϕ is some increasing smoothconcave function that is strictly sublinear Then (31) admits a unique invariant measuremicro and there exists a positive constant c such that for all x isin Rn the bound

Pt (x middot)minus microTV le cV (x)ψ(t)

holds where ψ(t) = 1(ϕ Hminus1ϕ )(t) and Hϕ(t) =

int t1 (1ϕ(s)) ds


33 Convergence speed toward the invariant measure lower bounds In order to obtainlower bounds on the rate of convergence towards the invariant measuremicro we are goingto make use of the following mechanism Suppose that we know of some function G thaton the one hand it has very heavy (non-integrable) tails under the invariant measure ofsome Markov process but on the other hand its moments do not grow too fast Then thisshould give a lower bound on the speed of convergence towards the invariant measuresince the moment bounds prevent the process from exploring its heavy tails too quicklyThis is made precise by the following elementary result

Theorem 36 Let Xt be a Markov process on a Polish space X with invariant measuremicro and let G X rarr [1infin) be such that

bull There exists a function f [1infin)rarr [0 1] such that the function Id middot f y rarr y f (y)is increasing to infinity and such that micro(G ge y) ge f (y) for every y ge 1

bull There exists a function g X timesR+ rarr [1infin) increasing in its second argument andsuch that E(G(Xt ) | X0 = x0) le g(x0 t)

Then one has the bound

microtn minus microTV ge 1

2f((Id middot f )minus1(2g(x0 tn))

) (32)

where microt is the law of Xt with initial condition x0 isin X

Proof It follows from the definition of the total variation distance and from Chebyshevrsquosinequality that for every t ge 0 and every y ge 1 one has the lower bound

microt minus microTV ge micro(G(x) ge y)minus microt (G(x) ge y) ge f (y)minus g(x0 t)

y

Choosing y to be the unique solution to the equation y f (y) = 2g(x0 t) the resultfollows 13

The problem is that in our case we do not in general have sufficiently good informa-tion on the tail behaviour of micro to be able to apply Theorem 36 as it stands Howeverit follows immediately from the proof that the bound (32) still holds for a subsequenceof times tn converging toinfin provided that the bound micro(G ge yn) ge f (yn) holds fora sequence yn converging to infinity This observation allows to obtain the followingcorollary that is of more use to us

Corollary 37 Let Xt be a Markov process on a Polish space X with invariant measuremicro and let W X rarr [1infin) be such that

intW (x) micro(dx) = infin Assume that there exist

F [1infin)rarr R and h [1infin)rarr R such that

bull h is decreasing andintinfin

1 h(s) ds ltinfinbull F middot h is increasing and limsrarrinfin F(s)h(s) = infinbull There exists a function g X timesR+ rarr R+ increasing in its second argument and such

that E((F W )(Xt ) | X0 = x0) le g(x0 t)

Then for every x0 isin X there exists a sequence of times tn increasing to infinity suchthat the bound

microtn minus microTV ge h((F middot h)minus1(g(x0 tn))

)

holds where microt is the law of Xt with initial condition x0 isin X

146 M Hairer

Proof Sinceint

W (x) micro(dx) = infin there exists a sequence wn increasing to infinitysuch that micro(W (x) ge wn) ge 2h(wn) for otherwise we would have the bound

intW (x) micro(dx) = 1 +

int infin

1micro(W (x) ge w) dw le 1 + 2

int infin

1h(w) dw ltinfin

thus leading to a contradiction Applying Theorem 36 with G = FW and f = 2hFminus1

concludes the proof 13

4 Existence and Non-existence of an Invariant Probability Measure

41 Non-existence of an invariant probability measure The aim of this section is toshow that (12) does not admit any invariant probability measure if k gt 2 or k = 2 andTinfin gt α2C Note first that one has an upper bound on the evolution of the total energyof the system given by

LH = γ T + γ Tinfin minus γ p20

which suggests that H is a natural choice for the function W2 in Wonhamrsquos criterion forthe non-existence of an invariant probability measure

It therefore remains to find a function W1 that grows to infinity in some direction (notnecessarily all) that is dominated by the energy in the sense that

limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)

and such that LW1 ge 0 outside of some compact region KIn order to construct W1 we use some of the ideas introduced in [HM08a] The

technique used there was to make a change of variables such that in the new variablesthe motion of the lsquofastrsquo oscillator decouples from that of the lsquoslow oscillatorrsquo In thesituation at hand we wish to show that the energy of the second oscillator grows so thatthe relevant regime is the one where that energy is very high

One is then tempted to set

W1 = Hminusζ (H minusH0) (42)

for some (typically small) exponent ζ isin (0 1) where H0 is a multiple of the energy ofthe first oscillator expressed in the lsquorightrsquo set of variables In order to compute LW1 wemake use of the following lsquochain rulersquo for L

L( f g) = (parti f g)Lgi + (part2i j f g)(gi g j ) (43)

(summation over repeated indices is implied) where we defined the lsquocarreacute du champrsquooperator

(gi g j ) = γ T partp0 gipartp0 g j + γ Tinfinpartp1 gipartp1 g j

(Note that it differs by a factor two from the usual definition in order to keep expressionsas compact as possible) This allows us to obtain the identity

LW1 = Hminusζ (γ T + γ Tinfin minus γ p20 minus LH0)

minusγ ζHminusζminus1(H minusH0)(T + Tinfin minus p20)

minus2γ ζHminusζminus1(T p0(p0 minus partp0H0) + Tinfin p1(p1 minus partp1H0))

+γ ζ(ζ + 1)Hminusζminus2(H minusH0)(

T p20 + Tinfin p2

1

) (44)


Following our heuristic calculation in Sect 21 we expect that at high energies one hasLH0 asymp γ T minusγ p2

0 where p0 denotes the correct variable in which to express the motionof the oscillator One would then like to first choose our compact set K sufficiently largeso that the expression on the first line of (44) is larger than δHminusζ (1+ p2

0) for some δ gt 0Then by choosing ζ sufficiently close to zero one would like to make the remainingterms sufficiently small so that LW1 gt 0 outside of a compact set This is made preciseby the following lemma

Lemma 41 Let L be as in (13) Assume that there exist a C2 function H0 R4 rarr Rand strictly positive constants c and C such that outside of some compact subset of R4it satisfies the bounds

LH0 le γ (T + Tinfin minus p20)minus c(1 + p2

0) |H0| + |partp0H0|2 + |partp1H0|2 le C H

If the function H0 furthermore satisfies

lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)

then (12) admits no invariant probability measure

Proof Setting W1 as in (42) we see from (44) and the assumptions on H0 that thereexists a constant C gt 0 independent of ζ isin (0 1) such that the bound

LW1 ge cHminusζ (1 + p20)minus ζC Hminusζ (1 + p2

0)

holds outside of some compact set Choosing ζ lt cC it follows that LW1 gt 0 outsideof some compact subset of R4 Assumption (45) makes sure that W1 grows to +infinin some direction and rules out the trivial choice H0 prop H Since it follows further-more from the assumptions that W1 le C H1minusζ (41) holds so that the assumptions ofWonhamrsquos criterion are satisfied 13

The remainder of this section is devoted to the construction of such a function H0thus giving rise to the following result

Theorem 42 There exists a constant C such that if either k gt 2 or k = 2 andTinfin gt α2C the model (12) admits no invariant probability measure

Remark 43 As will be seen from the construction the constant C is really equal to theconstant 〈2

p〉 from Sect 21

Proof As in [HM08a] we define the Hamiltonian

Hf (P Q) = P2

2+|Q|2k

2k

of a lsquofreersquo oscillator on R2 and its generator

L0 = PpartQ minus Q|Q|2kminus1partP (46)

These definitions will be used for all of the remainder of this article except for Sect 7The variables (P Q) should be thought of as lsquodummy variablesrsquo that will be replacedby for example (p1 q1) or (p0 q0) when needed

148 M Hairer

We also define as the unique centred1 solution to the Poisson equation

L0 = Q minusR(P Q)

where R R2 rarr R is a smooth function averaging out to zero on level sets of Hf andsuch that R = 0 outside of a compact set and R = Q inside an open set containingthe origin The reason for introducing the correction term R is so that the function is smooth everywhere including the origin which would not be the case otherwise It

follows from [HM08a Prop 37] that scales like H1kminus 1

2f in the sense that outside

a compact set it can be written as = H1kminus 1

2f 0(ω) where ω is the angle variable

conjugate to Hf Inspired by the formal calculation from Sect 21 we then define p0 = p0 minus

α(p1 q1) so that the equations of motion for the first oscillator turn into

dq0 = p0 dt + α dt

d p0 = minusq0|q0|2kminus2 dt minus αq0 dt minus γ p0 dt +radic

2γ T dw0(t) (47)

+αR dt minus α2(q0 minus q1)partP dt minus αradic2γ TinfinpartP dw1(t)minus αγ Tinfinpart2

P dt

+αRprime1(q1)partP dt minus Rprime1(q0) dt

Here we omitted the argument (p1 q1) from its partial derivatives and R in orderto make the expressions shorter Setting

H0 = p20

2+ Veff (q0) + θ p0q0 Veff (q) = V1(q) + α

q2

2 (48)

we obtain the following identity

LH0 = γ T minus (γ minus θ)p20 minus θ |q0|2k minus αθ |q0|2 minus γ θp0q0

+α2(γ minus θ)2 + α(γ minus θ) p0 + αV primeeff (q0)

+α p0Rminus α2 p0(q0 minus q1)partPminus αγ Tinfin p0part2P + α2γ Tinfin(partP)

2

+θq0

(αRminus α2(q0 minus q1)partPminus αγ Tinfinpart2

P)

+( p0 + θq0)αRprime1(q1)partP (49)

All the terms on lines 3 to 5 (and also the terms on line 2 provided that k gt 2) are of theform f (p0 q0)g(p1 q1) with g a function going to 0 at infinity and f a function suchthat f (p0 q0)Hf (p0 q0) goes to 0 at infinity It follows that for every ε gt 0 thereexists a compact set Kε sub R4 such that outside of Kε one has the inequality

LH0 le γ T minus γ p20 + ε +

(θ +

θγ 2

4α+ ε

)p2

0 minus (θ minus ε)|q0|2k

+α2(γ minus θ)2 + α(γ minus θ) p0 + αV primeeff (q0) (410)

1 We say that a function on R2 is centred if it averages to 0 along orbits of the Hamiltonian system withHamiltonian Hf


Here we also used the fact that γ θp0q0 le αθ |q0|2 + γ 2θ4α p2

0 If k gt 2 then the function also converges to 0 at infinity so that the bound


(θ +

θγ 2

4α+ ε

)p2


holds outside of a sufficiently large compact set It follows that the conditions ofLemma 41 are satisfied by H0 = (1 + δ)H0 for δ gt 0 sufficiently small wheneverTinfin gt 0 provided that one also chooses both θ and ε sufficiently small

The case k = 2 is slightly more subtle and we assume that k = 2 for the remainderof this proof In particular this implies that scales like a constant outside of somecompact set This suggests that the term 2 should average out to a constant whereasthe terms p0 andV primeeff (q0) should average out to zero modulo some lower-order cor-rections It turns out that these corrections will have the unfortunate property that theygrow faster than Hf in the (p0 q0) variables On the other hand we notice that bothp0 and V primeeff (q0) do grow slower than Hf at infinity As a consequence it is sufficient tocompensate these terms for lsquolowrsquo values of (p0 q0)

Before giving the precise expression for a function H0 that satisfies the assumptionsof Lemma 41 for the case k = 2 we make some preliminary calculations We denoteby ψ R rarr R+ a smooth decreasing lsquocutoff functionrsquo such that ψ(x) = 1 for x le 1and ψ(x) = 0 for x ge 2 Given a positive constant E we also set

ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)

Definition 44 We will say that a function f R+ times R4 rarr R is negligible if for everyε gt 0 there exists Eε gt 0 and for every E gt Eε there exists a compact set KEε R4

such that the bound | f (E p q)| le ε(1 + Hf ( p0 q0))

holds for every (p q) isin KEε

With this definition at hand we introduce the notations

f g f sim g (411)

to mean that there exists a negligible function h such that f le g + h or f = g + hrespectively With this notation we can rewrite (410) as

LH0 γ T minus γθ p20 minus θ |q0|2k + α2(γ minus θ)2 + fθ (412)

where we introduced the constant γθ = γ minus θ(1 + γ 2

4α ) and the function fθ = α(γ minusθ) p0 + αV primeeff (q0)

Lemma 45 Let a b ge 0 and let f g R2 rarr R be functions that scale like Haf and

Hminusbf respectively Then the following functions are negligible

i) f ( p0 q0)g(p1 q1)ψE ( p0 q0) provided that b gt 0ii) f ( p0 q0)g(p1 q1)ψ

primeE ( p0 q0) provided that b gt 0 or a lt 2

iirsquo) f ( p0 q0)g(p1 q1)(ψ primeE ( p0 q0)

)2 provided that b gt 0 or a lt 3

iii) f ( p0 q0)g(p1 q1)ψprimeprimeE ( p0 q0) provided that b gt 0 or a lt 3

iv) f ( p0 q0)g(p1 q1)(1minus ψE ( p0 q0)) provided that a lt 1

150 M Hairer

Proof We assume without loss of generality that the bounds f (p q) le 1 or Haf (p q)

and g(p q) le 1 and Hminusbf (p q) hold for every (p q) isin R2

In the case i) we take Eε = 1 and choose for KEε the set of points such that eitherHf ( p0 q0) ge 2E in which case the expression vanishes or Hf (p1 q1) ge (2E)abεminus1b

in which case the expression is smaller than εThe case ii) with b gt 0 follows exactly like the case i) so we consider the case a lt 2

and b = 0 Since ψ primeE = 0 if Hf ( p0 q0) ge 2E and is smaller than 1E otherwise wehave the bounds

| f ( p0 q0)g(p1 q1)ψprimeE ( p0 q0)| le (1 + Hf ( p0 q0))E

(0oraminus1)minus1

Since the exponent of E appearing in this expression is negative provided that a lt 2this is shown to be negligible by choosing Eε sufficiently large and setting KEε = φCases iirsquo) and iii) follow in a nearly identical manner

In the case iv) we use the fact that since a lt 1 for fixed ε gt 0 we can find aconstant Cε such that | f ( p0 q0)| le ε

2 Hf ( p0 q0)+Cε We then set Eε = 2Cεε so that

Hf ( p0 q0) ge Eε implies Hf ( p0 q0) ge 2Cεε

Since g is bounded by 1 by assumptionand since 1 minus ψE vanishes for Hf ( p0 q0) le E it follows that the expression iv) isuniformly bounded by εHf ( p0 q0) for E ge Eε 13Remark 46 In the case where both b gt 0 and a lt 1 the function f ( p0 q0)g(p1 q1)

is negligible which can be seen from cases i) and iv) above

Corollary 47 In the setting of Lemma 45 the following functions are negligible

v) f ( p0 q0)g(p1 q1)partp0ψE ( p0 q0) provided that b gt 0 or a lt 32vi) f ( p0 q0)g(p1 q1)partp1ψE ( p0 q0)

vii) f ( p0 q0)g(p1 q1)LψE ( p0 q0) provided that b gt 12 minus 1

k or b = 12 minus 1

k and a lt 1

Proof We can write

f ( p0 q0)g(p1 q1)partp0ψE ( p0 q0) = p0 f ( p0 q0)g(p1 q1)ψprimeE

f ( p0 q0)g(p1 q1)partp1ψE ( p0 q0) = p0 f ( p0 q0)g(p1 q1)partP(p1 q1)ψprimeE

so that the first two cases can be reduced to case ii) of Lemma 45 For case vii) we usethe fact that

LψE = ψ primeELHf + γ(

T0 + Tinfin(partP)2)

p20ψprimeprimeE (413)

and that LHf consists of terms that all scale like Hcf ( p0 q0)Hd

f (p1 q1) with c le 1 and

d le 1k minus 1

2 (see (49)) to reduce ourselves to cases ii) and iii) of Lemma 45 13Before we proceed with the proof of Theorem 42 we state two further preliminary

results that will turn out to be useful also for the analysis of the case k isin (1 2)

Lemma 48 Let k isin (1 2] and let f R2 rarr R be a function that scales like Haf for

some a isin R Then the function g = L( f ( p0 q0)) consists of terms that are bounded bymultiples of Hc

f ( p0 q0)Hdf (p1 q1) with either c le a + 1

2 minus 12k and d le 0 or c le aminus 1

2k

and d le 1k minus 1

2


Proof It follows from (47) that

g =(minusαq0 minus γ ( p0 + α) + αRminus α2(q0 minus q1)partPminus αγ Tinfinpart2

P)partP f

+(αRprime1(q1)partPminus Rprime1(q0)

)partP f

+γ(

T + Tinfin(partP)2)part2

P f + ( p0 + α)partQ f minus q0|q0|2kminus2partP f

from which the claim follows by simple powercounting 13

Lemma 49 Let k isin (1 2] and let f R2 rarr R be a function that scales like Hminusbf for

some b isin R Then the function g = L( f (p1 q1))minus(L0 f )(p1 q1) consists of terms thatare bounded by multiples of Hc

f ( p0 q0)Hdf (p1 q1) with either c le 1

2k and d le minusbminus 12

or c le 0 and d le minusb minus 12 + 1

2k


g = α(q0 minus q1)partP f minus Rprime1(q1)partP f + γ Tinfinpart2P f (414)

from which the claim follows 13We now return to the proof of Theorem 42 We define13 as the unique centred solu-

tion to the equation L013 = One can see in a similar way as before that13 scales like

Hminus 1

4f Since scales like a constant there exists some constant C such that2 averages

to C outside a compact set While the constant C can not be expressed in simple termsit is easy to compute it numerically C asymp 063546992

In particular there exists a function R R+ rarr R+ with compact support and suchthat2minus C +R(Hf (P Q)) is centred Denote by the centred solution to the equation

L0 = 2 minus C + R(Hf (P Q)) (415)

so that scales like Hminus 1

4f just like 13 does With these definitions at hand we set

H0 = H0 minus(α2(γ minus θ)(p1 q1) + fθ13(p1 q1)

)ψE ( p0 q0) (416)

where we used the function fθ introduced in (412) Recalling that fθ consists of termsscaling like Ha

f ( p0 q0) with a le 34 we obtain from Lemmas 49 and 45 that

fθL(13(p1 q1))ψE = fθminus fθ(1minus ψE ) + fθ (L13 minus L013)ψE sim fθ

Similarly we obtain that

L((p1 q1))ψE = 2 minus C minus(2 minus C

)(1minus ψE ) + RψE sim 2 minus C

2 All displayed digits are accurate

152 M Hairer

It therefore follows from (412) the facts that partp0 p0 = 1 and partp1 p0 = minusαpartP(p1 q1)and the multiplication rule for L that one has the bound

LH0 γ T minus γθ p20 minus θ |q0|2k + α2(γ minus θ)C

minus(α2(γ minus θ) + fθ13)LψE minus L fθ13ψE

+C |partPpartp1ψE | + C | fθ partP13partp1ψE |+C |13partP fθ

(1 + α2(partP)

2)ψ primeE | + C |partP13partP fθ partPψE |

The terms on the second and third line are negligible by Lemma 48 and Corollary 47The terms on the last line are negligible by Lemma 45 so that we finally obtain thebound

LH0 γ (T + α2C)minus γθ p20 minus θ |q0|2k minus α2θC (417)

Since the constant γθ can be made arbitrarily close to γ by choosing θ sufficiently smallwe see as before that provided that Tinfin gt α2C it is possible to choose θ small enoughand E large enough so that the choice H0 = (1+δ)H0 with δ gt 0 sufficiently small againallows to satisfy the conditions of Lemma 41 This concludes the proof of Theorem 42

13

42 Existence of an invariant measure Theorem 42 has the following converse

Theorem 410 If either 1 lt k lt 2 or k = 2 and Tinfin lt α2C the model (12) admits aunique invariant probability measure micro The constant C is the same as in Theorem 42

Proof Somewhat surprisingly given that the two statements are almost diametricallyopposite it is possible to prove this positive result in very similar way to the previousnegative result by constructing the right kind of Lyapunov function As before the casek = 2 will be treated somewhat differentlyThe case k = 2 Similarly to what we did in (42) the idea is to look at the functionV = H minus cH0 for a suitable constant c but this time we choose it in such a way thatlim|(pq)|rarrinfin V = infin and lim sup|(pq)|rarrinfin LV lt 0 so that we can apply Theorem 31Note that with the same notations as in the proof of Theorem 42 one has from (49)

LH0 sim γ T minus (γ minus θ)p20 minus θ |q0|2k minus αθ |q0|2 minus γ θp0q0

+α2(γ minus θ)2 + fθ

so that provided this time that we choose θ lt 0 in the definition of H0 (and thereforeof H0) we have the bound

LH0 sim γ T + α2(γ minus θ)C minus (γ minus θ)p20 minus θ |q0|2k minus αθ |q0|2 minus γ θp0q0

γ T + α2(γ minus θ)C minus γθ p20 minus θ |q0|2k

where we set γθ = γ minus θ(1 + γ 2

4α ) as before Here the function H0 is as in (416) anddepends on a large parameter E as above If we choose c lt 1 the function

V = H minus cH0 (418)


does then indeed grow to infinity in all directions and we have

LV γ T (1minus c) + γ Tinfin minus cα2(γ minus θ)C minus c|θ ||q0|2k minus (γ minus cγθ )|p0|2If the assumption α2C gt Tinfin is satisfied we can find a constant β gt 0 such that

γ T (1minus c) + γ Tinfin minus cα2(γ minus θ)C le minusβfor all θ sufficiently small and all c sufficiently close to 1 By fixing c and making θsufficiently small we can furthermore ensure that γ minus cγθ gt 0 This shows that by firstchoosing c sufficiently close to 1 then making θ very small and finally choosing E verylarge we have constructed a function V satisfying the assumptions of Theorem 31 thusconcluding the proof in the case k = 2The case k lt 2 Even though one would expect this to be the easier case it turns out tobe tricky because of the fact that the approximate decoupling of the oscillators at highenergies is not such a good description of the dynamic anymore The idea is to consideragain the variable p0 introduced previously but because of the fact that the function is now no longer bounded we are going to multiply certain correction terms by a lsquocutofffunctionrsquo

Since we are following a similar line of proof to the non-existence result and sincewe expect from (25) and (26) to be able to find a function V close to H and such that it

asymptotically satisfies a bound of the type LV asymp minusHf ( p0 q0) minus H2kminus1

f (p1 q1) thissuggests that we should introduce the following notion of a negligible function suited tothis particular case

Definition 411 A function f R4 rarr R is negligible if for every ε gt 0 there exists a

compact set Kε such that the bound | f (p q)| le ε(

Hf ( p0 q0) + H2kminus1

f (p1 q1)

)holds

for every (p q) isin Kε

We also introduce the notations sim and similarly to before For θ gt 0 we then setV = H + θ p0q0 so that (47) yields

LV = γ (T + Tinfin)minus γ p20 + αθ p0 + θ p2

0 minus θ |q0|2k minus αθ |q0|2 minus γ θp0q0

+θαq0

(Rminus α(q0 minus q1)partPminus γ Tinfinpart2

P)

+θq0(αRprime1(q1)partPminus Rprime1(q0)

)

It is straightforward to check that all of the terms on the second and third lines are negli-gible Using the definition of p0 and completing the square for the term α|q0|2 + γ p0q0we thus obtain the bound

LV minusγθ p20 minus θ |q0|2k + cθ p0minus αθ2 (419)

Here we defined the constants

αθdef= αγ

(α minus γ θ

4

) cθ

def= (αθ minus 2αγ + 12γ

2θ)

in order to shorten the expressions

154 M Hairer

As before we see that there exists a positive constant C and a function R R2 rarr R+

with compact support such that 2 minus C H2kminus1

f + R is smooth centred and vanishes ina neighbourhood of the origin Similarly to (415) we define as the unique centredsolution to

L0(P Q) = 2(P Q)minus C H2kminus1

f (P Q) + R(P Q)

and 13 as the unique centred solution to L013 = Note that scales like H5

2kminus 32

f and

that 13 scales like H3

2kminus1f

At this stage we would like to define V = V + αθ(p1 q1) minus cθ p013(p1 q1) inorder to compensate for the last two terms in (419) The problem is that when apply-ing the generator to p013 we obtain an unwanted term of the type q0|q0|2kminus213 whichgrows too fast in the q0 direction We note however that the term p0 only needs tobe compensated when | p0| which is the regime in which the description (47) isexpected to be relevant We therefore consider the same cutoff function ψ as before andwe set

V = V + αθ(p1 q1)minus cθ p013(p1 q1)ψ

(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)

for a positive exponent η to be determined laterIn order to obtain bounds on LV we make use of the fact that Lemma 49 still applies

to the present situation In particular we can apply it to the function thus obtainingthe bound

LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))

for some constant Cθ where it is understood that the function ψ is composed with theratio appearing in (420) Using the fact that L013 = by definition and applying thechain rule (43) for L we thus obtain

LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)

minuscθ ((L p0)13ψ + p013Lψ + p0(Lminus L0)13 + p0L13(ψ minus 1)))

minuscθT(partp1 p0partP13ψ + partp1 p013partp1ψ + p0partP13partp1ψ

)minus cθTinfin13partp0ψ

(421)

We claim that all the terms appearing on the second and the third line of this expressionare negligible thus concluding the proof The most tricky part of showing this is toobtain bounds on Lψ

Define E0 = 1 + Hf ( p0 q0) and E1 = 1 + Hf (p1 q1) as a shorthand Our main toolin bounding LV is then the following result which shows that the terms containing Lψare negligible

Proposition 412 Provided that η isin [2minus k k] there exists a constant C such that∣∣∣∣Lψ

(E0

Eη1

)∣∣∣∣ le C

∣∣∣∣partp0ψ

(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1


Proof Define the function f R2+ rarr R+ by f (x y) = ψ((1 + x)(1 + y)η) It can then

be checked by induction that for every pair of positive integers m and n with m + n gt 0and for every real number β there exists a constant C such that the bound

|partmx part

ny f | le (1 + x)minusm+β(1 + y)minusnminusηβ (422)

holds uniformly in x and y It furthermore follows from (47) and (12) that

|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)

|partp0 E0| le C E120 |partp0 E1| = 0

|partp1 E0| le C E120 E

1kminus11 |partp1 E1| le C E

121

Combining these two bounds with (422) and the chain rule (43) the required boundsfollow 13

Let us now return to the bound on LV It is straightforward to check that

|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)

for some constant C so that

|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)

which is negligible Combining Proposition 412 with the scaling behaviours of and13one can check in a similar way that the term p013Lψ as well as all the terms appearingon the third line of (421) are also negligible It therefore remains to bound p0(LminusL0)13

and p0L13(ψ minus 1) It follows from (414) that

|(Lminus L0)13| le C E3

2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)

so that | p0(LminusL0)13| is negligible as well Since we know that L013 scales like E1kminus 1

21

it follows from (423) that

| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)

This term has of course no chance of being negligible we have to use the fact that it ismultiplied by 1minusψ The function 1minusψ is non-vanishing only when E0 ge Eη1 so thatwe obtain

| p0L13(1minus ψ)| le C

(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer

We see that both exponents are strictly smaller than 1 provided that η gt 2k minus 1 Com-

bining all of these estimates with (421) we see that provided that η isin ( 2k minus 1 k) there

exists a constant C such that

LV minusC

(E0 + E

2kminus11

)

In particular using the scaling of we deduce the existence of a constant c such thatthe bound

LV le minuscV2kminus1 (424)

holds outside of a sufficiently large compact set (we can choose such a set so that V ispositive outside) thus concluding the proof of Theorem 410 by applying Theorem 31

13

5 Integrability Properties of the Invariant Measure

The aim of this section is to explore the integrability properties of the invariant measuremicro when it exists First of all we show the completely unsurprising fact that

Proposition 51 For all ranges of parameters for which there exists an invariant mea-sure micro one has

intexp (βH(x)) micro(dx) = infin for every β gt 1T

Proof Choose β gt β2 gt 1T Setting W2(x) = exp(β2 H(x)) we have

LW2 = γβ2W2

(T + Tinfin minus p2

0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)

outside of a sufficiently large compact set Setting similarly W1 = exp(H(x)T ) wesee immediately from a similar calculation that LW1 ge 0 so that the result follows fromTheorem 32 13Remark 52 Actually one can show similarly a slightly stronger result namely that thereexists some exponent α lt 1 such that Hα exp(HT ) is not integrable against micro

51 Energy of the first oscillator What is maybe slightly more surprising is that thetail behaviour of the distribution of the energy of the first oscillator is not very stronglyinfluenced by the presence of an infinite-temperature heat bath just next to it providedthat we look at the correct set of variables Indeed we have

Proposition 53 Let either 32 le k lt 2 or k = 2 and Tinfin be such that there exists an

invariant probability measure micro Thenint

exp(βHf ( p0 q0)

)micro(dx) lt infin for every

β lt 1T

Remark 54 When k = 2 is bounded and the exponential integrability of Hf ( p0 q0)

is equivalent to that of Hf (p0 q0) This is however not the case when k lt 2

Remark 55 The borderline case k = 32 is expected to be optimal if we restrict ourselves

to the variables ( p0 q0) This is because for k lt 32 one would have to add additional

correction terms taking into account the nonlinearity of the pinning potential


The main ingredient in the proof of Proposition 53 is the following propositionwhich is also going to be very useful for the non-integrability results later in this section

Proposition 56 For every θ gt 0 there exist functions H0 p0 R4 rarr R and a constantCθ such that

bull For every ε gt 0 there exists a constant Cε such that the bounds

0 le H0 le (1 + ε)H + Cε (51)

holdbull Provided that k ge 3

2 for every ε gt 0 there exists a constant Cε such that the bound

(1minus ε)Hf ( p0 q0)minus Cε le H0 le (1 + ε)Hf ( p0 q0) + Cε (52)

holdsbull One has the bounds

(partp0 H0 minus p0)2 le Cθ + θ4H0 (53a)

(partp1 H0)2 le Cθ + θ4H0 (53b)

bull If furthermore k ge 32 the bound LH0 le Cθ minus (γ minus 2θ) p2

0 minus θH0 holdsbull If k isin (43 32) then for every δ gt (2k minus 1)( 3

k minus 2) one has the bound LH0 leCθ minus (γ minus 2θ) p2

0 minus θH0 + θ2 H δf (p1 q1)

Remark 57 The presence of p0 rather than p0 in (52) is not a typographical mistake

Proof We start by defining the differential operator K acting on functions F R2 rarr Ras

KF = γ Tinfin(part2

P F)(p1 q1) +

(α(q0 minusq(p1 q1)minus q1

)minus Rprime1(q1))(partP F)(p1 q1)

so that KF = L(F(p1 q1))minus (L0 F)(p1 q1) Setting

p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)

for some yet to be defined functionsp andq and for Ei andψ as in Proposition 412we then obtain

dq0 = p0 dt +(L0q minusp

)dt

+ψKq dt + (ψ minus 1)L0q dt +qLψ dt + γ Tinfinpartp1ψpartPq dt

+radic

2γ Tinfin(ψpartPq +qpartp1ψ

)dw1(t) +

radic2γ Tqpartp0ψ dw0(t)

d p0 = minusV primeeff (q0) dt minus γ p0 dt +radic

2γ T dw0

+(L0p minus αq1 + γp

)dt

+Kp dt +(

V primeeff (q0)minus V primeeff (q0))

dt +radic

2γ TinfinpartPp dw1(t) (55)

where we defined as before the effective potential Veff (q) = V1(q) + α q2

2

Let E gt 0 and set (1)p as the unique centred solution to

L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer

where ψ is the same cutoff function already used previously We then define (2)p by

L0(2)p = γ(1)p and we set p = (1)p +(2)p This ensures that one has the identity

L0p minus αq1 + γp = Rp

where the function Rp consists of terms that scale like Haf with a le 3

2k minus1 We further-more set q to be the unique centred solution to L0q = p Note that p consistsof terms scaling like Ha

f with a le 1k minus 1

2 and that q consists of terms scaling like Haf

with a le 32k minus 1 The introduction of the parameter E in (56) ensures that we can make

functions scaling like a negative power of Hf arbitrarily small in the supremum norm

It follows indeed that one has for example |partPp| le C E1kminus1

With these definitions at hand it follows from (55) that

dq0 = p0 dt +radic


)dw1(t) +


+ψKq dt + (ψ minus 1)p dt +qLψ dt + γ Tinfinpartp1ψpartPq dt


2γ T dw0

+Rp dt + Kp dt +(


dt +radic

2γ TinfinpartPp dw1(t)

(57)

Let now H0 be defined by

H0 = p20

2+ Veff (q0) + θ p0q0 + C0

where C0 is a sufficiently large constant so that H0 ge 1 Note that as a consequence ofthe definitions of p0 and q0 if k ge 32 then | p0minus p0| and |q0minus q0| are bounded so thatthe two-sided bound (52) does indeed hold Showing that the weaker one-sided bound(51) holds for every k isin [ 32 2] is straightforward to check

Before we turn to the proof of (53) let us recall the definitions of E0 and E1 fromthe proof of the case k lt 2 of Theorem 410 and define similarly E0 = 1 + Hf ( p0 q0)If k ge 3

2 then E0 and E0 are equivalent in the sense that they are bounded by multiplesof each other If k lt 3

2 this is not the case but it follows from the definitions ofp andq that

E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)

It follows that provided that we impose the condition η gt 3 minus 2k where η is theexponent appearing in (54) then one has the implications

E0 le C Eη1 rArr E0 le C Eη1 (58a)

E0 ge C Eη1 rArr E0 ge C Eη1 (58b)

for some constant C depending on C We will assume from now on that the conditionη gt 3minus 2k is indeed satisfied Let us now show that (53a) holds We have the identity

partp0 H0 minus p0 = θ q0 +(

V primeeff (q0) + θ p0

)qpartp0ψ

Since the term θ q0 satisfies the required bound we only need to worry about the sec-ond term It follows from Proposition 412 and from the scaling of q that this term is


bounded by a multiple of E1minus 1

2k0 E

32kminus1minus η21 Since the bounds (58) hold on the support

of partp0ψ this in turn is bounded by a multiple of E120 E

32kminus1minus η

2k1 so that the requested

bound follows provided again that the condition η gt 3minus 2k holdsTurning to (53b) we have the identity

partp1 H0 = ( p0 + θ q0)partPp +(


)partp1(qψ)

Making use of the parameter E introduced in (56) it follows that the first term is

bounded by E120 E

1kminus1 which can be made sufficiently small by choosing E θ

2k1minusk

In order to bound the second term we expand the last factor into qpartp1ψ + ψpartPq The first term can be bounded just as we did for partp0 H0 noting that the bound on partp1ψ

in Proposition 412 is better than the bound on partp0ψ Using the fact that (58a) holds

on the support of ψ the second term yields a bound of the form E120 E

ηminus32 (1minus 1

k )

1 whichyields the required bound provided that η lt 3

It therefore remains to show the bound on LH0 It follows from (57) that one hasthe identity

LH0 = γ T minus (γ minus θ) p20 minus θ |q0|2k minus αθ |q0|2 minus γ θ p0q0

+γ Tinfin((partPp)

2 + V primeprimeeff (q0)(partp1(qψ)

)2 + 2θpartPppartp1(qψ))

+γ T(

V primeprimeeff (q0)(qpartp0ψ)2 + 2θqpartp0ψ

)

+( p0 + θ q0)(Rp + Kp + V primeeff (q0)minus V primeeff (q0)

)+

(V primeeff (q0) + θ p0

)

times (ψKq + (ψ minus 1)p +qLψ + γ Tinfinpartp1ψpartPq

) (59)

We now use the following notion of a negligible function A function f R+timesR4 rarr Ris negligible if for every ε gt 0 there exists a constant Eε and for every E gt Eε there

exists a constant Cε such that the bound | f (E p q)| le Cε + ε(

E0 + Eδ1

)holds where

δ is as in the statement of the proposition (Set δ = 0 for k ge 32)With this notation the required bounds follow if we can show that all the terms

appearing in (59) are negligible except for those on the first line The terms appearingin the second line are all smaller than the last term appearing in partp1 H0 and so theyare negligible Similarly the terms appearing in the third line are smaller than thoseappearing in partp1 H0 minus p0

It is easy to see that the first term on the fourth line is negligible Concerning the

second term we see that |Kp| le C

(E

12k0 + E

32kminus11

) so that this term is also seen to

be negligible by power counting Note now that the definitions of Veff and q0 imply thatone has the bound

∣∣∣V primeeff (q0)minus V primeeff (q0)

∣∣∣ le C

(1 + |q0|2kminus2 + |q0|2kminus2

)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E

(2kminus2)( 32kminus1)

1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer

Furthermore one has V primeeff (q0) = V primeeff (q0) unless E0 le Eη1 so that we have the bound

p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)

The second term is always negligible Furthermore if η gt (3 minus 2k)(2 minus k) the firstterm is also negligible

We now turn to the last line in (59) In order to bound the term involving Kq note that the functions qpartPq part2

Pq and QpartPϕq are bounded provided that k ge43 so that the terms involving these expressions are negligible Concerning the termV primeeff (q0)q0partPq we use the fact that partPq can be made arbitrarily small by choosingE large enough in (56) to conclude that it is also negligible The term involving p is


2k + 1η( 1

kminus 12 )

0 so that it is negligible provided that η gt 2minus kThe term involving qLψ is bounded similarly using the fact that Lψ is bounded byProposition 412 and that q scales like a smaller power of Hf than p Finally thelast term is negligible since partp1ψpartPq is bounded thus concluding the proof of Prop-osition 56 Note that the choice η = 2 for example allows to satisfy all the conditionsthat we had to impose on η in the interval k isin [43 2] 13

We are now able to give the

Proof of Proposition 53 It follows from (52) that if we can show that exp(βH0) isintegrable with respect to micro for every β lt 1T then the same is also true forexp(βHf ( p0 q0)) provided that we restrict ourselves to the range k ge 3

2 Before we proceed we also note that (53a) implies that for θ sufficiently small one

has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0

for some constant Cθ Setting W = exp(βH0) we thus have the bound

LW

βW= LH0 + γβ

(T (partp0 H0)

2 + Tinfin(partp1 H0)2)

le Cθ minus (γ minus 2θ) p20 minus θH0 + γβ(1 + θ) p2

0 + Cθ2H0

for some constant C independent of θ Since we assumed that β lt 1T we can makeθ sufficiently small so that minus(γ minus 2θ) + γβ(1 + θ) lt 0 and Cθ2 minus θ lt 0 The claimthen follows from Theorem 31 13

52 Integrability and non-integrability in the case k = 2 We next show that if k = 2and Tinfin le α2C then the invariant measure is heavy-tailed in the sense that there existsan exponent ζ such that

intH ζ (x) micro(dx) = infin Our precise result is given by

Theorem 58 If k = 2 and Tinfin le α2C one hasint

H ζ (x) micro(dx) = infin provided that

ζ gt ζdef= 3

4


Conversely one hasint

H ζ (x) micro(dx) ltinfin for ζ lt ζ


Proof We first show the positive result namely that H ζ is integrable with respect to microfor any ζ lt ζ Fixing such a ζ our aim is to construct a smooth function W boundedfrom below such that for some small value ε gt 0 the bound LW le minusεH ζ holdsoutside of some compact set This then immediately implies the required integrabilityby Theorem 31

Consider the function V defined in (418) Note that this function depends on param-eters E θ and c and that for any given value of ε gt 0 it is possible to choose first θsufficiently small and c sufficiently close to 1 and then E sufficiently large so that thebound

LV le γ Tinfin minus α2γ C + ε

holds outside of some compact setLet us now turn to the behaviour of partp0 V and partp1 V It follows from the definitions

Lemma 45 and Corollary 47 that one has the identity

(partp0 V

)2 = (1minus c)2 p20 + R0

where the function R0 can be bounded by an arbitrarily small multiple of V outside ofsome sufficiently large compact set Furthermore it follows from the definition of Vand the construction of H0 that one has the bound V ge 1minusc

2 H outside of some compactset so that we have the bound

(partp0 V

)2 le 4(1minus c)V + R0

Ensuring first that 1 minus c le ε8 and then choosing E sufficiently large it follows thatwe can ensure that

(partp0 V

)2 le εV outside of a sufficiently large compact set It followsin a similar way that by possibly choosing E even larger the bound

(partp1 V

)2 le p21 + εV

holds outside of some compact set Note now that since

L0(P Q) = 3P2 minus 4Hf (510)

the function P2minus 43 Hf is centred Let furthermore R R2 rarr R be a centred compactly

supported function such that P2 minus 43 Hf + R vanishes in a neighbourhood of the origin

and let be the centred solution to

L0 = P2 minus 4

3Hf + R (511)

so that we have the identity

L(p1 q1) = p21 minus

4

3Hf (p1 q1) + R(p1 q1) +

(α(q0 minus q1)minus Rprime1(q1)

)(partP

)(p1 q1)

Furthermore it follows at once from the definition of V and the scaling behaviours of and 13 that the bound

Hf (p1 q1) le (1 + ε)V

162 M Hairer

holds outside of some compact set Since furthermore R is bounded and scales like

H34

f it follows that the bound

L(p1 q1) ge p21 minus

4

3(1 + ε)V

holds outside of some (possibly larger) compact set Finally it follows from the scalingof that the bounds

|LV | le εV and |partp1 V partp1| le εV (512)

hold outside of some sufficiently large compact setWith all these definitions at hand we consider the function

W = V ζ+1 minus γ ζ(ζ + 1)TinfinV ζ (p1 q1) (513)

Note that V is positive outside of a compact set so that W is well-defined there Since wedo not care about compactly supported modifications of W we can assume that (513)makes sense globally We then have the identity

LW = (ζ + 1)V ζLV + ζγ (ζ + 1)V ζminus1(

T(partp0 V

)2 + Tinfin(partp1 V

)2 minus TinfinL)

minus γ ζ 2(ζ + 1)TinfinV ζminus1(LV + γ Tinfinpartp1 V partp1

)

Collecting all of the bounds obtained above this in turn yields the bound

LW le (ζ + 1)V ζ(γ Tinfin minus α2γ C + ε

)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)

minus γ εζ 2(ζ + 1)TinfinV ζ

le γ (ζ + 1)

(Tinfin minus α2C +

4

3ζTinfin + K ε

)V ζ

holding for some constant K gt 0 independent of ε outside of some sufficiently largecompact set It follows that if ζ lt ζ it is possible to choose ε sufficiently small so thatthe prefactor in this expression is negative thus yielding the desired result

We now prove the lsquonegative resultrsquo namely that H ζ is not integrable with respectto micro if ζ gt ζ In order to show this we are going to apply Wonhamrsquos criterion withW2 = H1+ζ It therefore suffices to find a function W1 growing to infinity in somedirection such that LW1 gt 0 outside of some compact set and such that

supH(pq)=E

W1(p q)Eminus1minusζ rarr 0 (514)

as E rarr infin We are going to construct W1 in a way very similar to the construction inthe proof of the positive result above

Fix some arbitrarily small ε gt 0 as before Setting V as above note first that it followsimmediately from (417) that by choosing first θ sufficiently small then c sufficientlyclose to 1 and finally E large enough we can ensure that the bound

LV ge γ Tinfin minus α2γ C minus ε(1 + p20)


holds outside of some sufficiently large compact set Similarly as before we can alsoensure that the bound

(partp1 V

)2 le p21 minus εV

holds Fix now some ζ isin (ζ ζ ) and define W0 as in (513) but with ζ replacing ζ Itfollows that the bound

LW0 ge γ (ζ + 1)


4

3ζTinfin minus K ε(1 + p2

0)

)V ζ

holds for some constant K gt 0 outside of some compact set The problem is that theright hand side of this expression is not everywhere positive because of the appearanceof the term p2

0 This can however be dealt with by setting

W1 = W0 minus K εH1+ζ (515)

so that

LW1 ge γ (ζ + 1)


4

3ζTinfin minus K ε

)V ζ

for some different constant K Since ζ gt ζ we can ensure that this term is uniformlypositive by choosing ε sufficiently small By possibly making ε even smaller we canfurthermore guarantee that W1 grows in some direction despite the presence of thetermminusK εH1+ζ in (515) Finally the condition (514) is guaranteed to hold because wechoose ζ lt ζ 13

As a corollary of Theorem 58 we obtain

Corollary 59 If k = 2 and α2C gt Tinfin gt 37α

2C then even though the system admitsa unique invariant measure micro the average kinetic energy of the second oscillator isinfinite that is

intp2

1 micro(dx) = infin

Proof The proof is very similar to the proof of the ldquonegative partrdquo of Theorem 58However instead of choosing W2 = H2 we choose W2 = H2 + K p1q1 for some con-stant K Since this additional term does not change the behaviour of W2 at infinity theconclusions of Wonhamrsquos criterion still apply showing that (LW2)+ is not integrablewith respect to micro A simple explicit calculation shows that provided that K is largeenough there exists a positive constant C such that LW2 le C

(1 + Hf (p0 q0) + p2

1

)

On the other hand we know that the expectation of Hf (p0 q0) is finite under micro bythe remark following Proposition 53 so that the expectation of p2

1 under micro necessarilydiverges 13

53 Integrability and non-integrability in the case k lt 2 In this case we show that theexponential of a suitable fractional power of H is integrable with respect to the invariantmeasure Our positive result is given by

Theorem 510 For every k isin (1 2) there exists δ gt 0 such thatint

R4exp

(δH

2kminus1(x)

)micro(dx) ltinfin (516)

where micro is the unique invariant measure for (12)

164 M Hairer

Proof Define W = exp(δV κ) for a (small) constant δ gt 0 and an exponent κ isin (0 1]to be determined later (the optimal exponent will turn out to be κ = 2

k minus 1) Here V isthe function that was previously defined in (420) Since V and H are equivalent in thesense that there exist positive constants C1 and C2 such that

Cminus11 V minus C2 le H le C1V + C2

showing the integrability of W implies (516) for a possibly different constant δApplying the chain rule (43) we obtain outside of a sufficiently large compact set

the bound

LW = δκW(

V κminus1LV + (δκV 2κminus2 + (κ minus 1)V κminus2)(V V ))

le δκW V κminus1(LV + 2δκV κminus1(V V )

) (517)

Note now that it follows immediately from (420) and Proposition 412 that outside ofsome compact set one has the bounds

|partp0 V | le C

(E

120 + E

120 E

32kminus1minus α21 + E

32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V

so that(V V ) le CV Combining this with (424) we obtain the existence of constantsc and C (possibly depending on κ but not depending on δ) such that

LW le δW V κminus1(

C + CδV κ minus cV2kminus1

) (518)

thus concluding the proof 13We have the following partial converse to Theorem 510

Theorem 511 Let k isin ( 43 2) Then there exists gt 0 such that

int

R4exp

(H

2kminus1(x)

)micro(dx) = infin (519)


Proof We are again going to make use of Wonhamrsquos criterion Let K be a (suffi-ciently large) constant define κ = 2

k minus 1 isin (0 12 ) set F(x) = exp(Hκ(x)) and

set W2(x) = exp(

K Hκ(x))

We then have the bound

LW2

γ κ K W2= Hκminus1


0

)+ Hκminus2(κ minus 1 + κ K Hκ)

(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)

for some constant C gt 0 In particular we have LV le F outside of some compact setprovided that we choose gt K


Similarly to 511 we denote by the centred solution to

L0 = Hf minus k + 1

2kP2 + R

for some compactly supported function R ensuring that the right hand side vanishes ina neighbourhood of the origin

Let now K be any constant smaller than K let M be a (large) positive constant to bedetermined later and set

W1 = exp(

K(

Hκ minus 2Hκminus1H0 + M H2κminus2(p1 q1)))

def= exp(K H1)

where H0 is the function from Proposition 56 and was defined above Note that theproperties of H0 imply that outside of some compact set one has the bounds

1 le H0 le (1 + ε)H

It is clear that W2 is much larger than W1 at infinity so that it remains to show thatLW1 gt 0 outside of a compact set for K sufficiently large We are actually going toshow that there exists a constant C such that (LW1)W1 ge C H2κminus1 outside of somecompact set Therefore we call a function f negligible if for every ε gt 0 there exists acompact set such that | f | le εH2κminus1 outside of this set Note that since we consider therange of parameters such that κ lt 1

2 bounded functions are not negligible in generalUsing the chain rule (43) we have the identity

LW1

K W1= LH1 + γ K

(T (partp0 H1)


ge LH1 + γ K Tinfin(partp1 H1)2

We first turn to the estimate of LH1 Using again (43) we have the identity

LH1 =(κHκminus1 + 2(1minus κ)Hκminus2H0

)LH minus 2Hκminus1LH0 + M H2κminus2L

+ 2(κ minus 1)M H2κminus3LH + γM(2κ minus 2)(2κ minus 3)H2κminus4(

T p20 + Tinfin p2

1

)

+ γ (κ minus 1)Hκminus3(κH + 2(2minus κ)H0)(

T p20 + Tinfin p2

1

)

+ 2γ (1minus κ)Hκminus2(T p0partp0 H0 + Tinfin p1partp1 H0)

+ γMTinfin(2κ minus 2)H2κminus3 p1partP

We see immediately that since κ is strictly positive and since scales like a power ofthe energy strictly smaller than one all terms except for the ones on the first line arenegligible Furthermore it follows from (51) that

κHκminus1 + 2(1minus κ)Hκminus2H0 le(

2minus κ2

)Hκminus1

say Combining this with Proposition 56 and the fact that the inequality κ gt (2k minus1)( 3

k minus 2) holds in the range of parameters under consideration we obtain the lowerbound

LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer

Using the definition of p0 and choosing θ lt γ κ8 we obtain the existence of a constantC such that

LH1 Hκminus1(

2θH0 minus C Hκf (p1 q1)

)+ M H2κminus2L0

Here we also made use of the scaling properties of in order to replace L by L0Note that the constant C appearing in the expression above can be made independentof θ provided that we restrict ourselves to θ le γ κ16 say At this point we make thechoice M = 2C and we set c = k+1

2k so that we have the lower bound

LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)

)+ M H2κminus2(Hf (p1 q1)minus cp2

1)

minusM

2Hκminus1 Hκ

f (p1 q1) + M H2κminus2(H0 + Hf (p1 q1)minus cp21)

where we made use of the fact that since κ lt 1 for every constant C there is a compactset such that Hκminus1H0 ge C H2κminus2H0 outside of that compact set From the definitionsof H and H0 we see that there exists a constant C and a compact set outside of whichCH0 + Hf (p1 q1) gt

34 H say so that we finally obtain the lower bound

LH1 M

4H2κminus1 minus cM H2κminus2 p2

1 (520)

Let us now turn to the term (partp1 H1)2 We have the identity

partp1 H1 = Hκminus2(κH + 2(1minus κ)H0)p1 + 2M(κ minus 1)H2κminus3p1

minus 2Hκminus1partp1 H0 + M H2κminus2partP

Using the inequality (a + b)2 ge a2

2 minusb2 as well as the bound (53b) it follows that thereexists a constant C such that the bound

(partp1 H1)2 ge κ2

2H2κminus2 p2

1 minus 16θ4 H2κminus2H0 minus C H2κminus2

holds Combining this bound with (520) we obtain the lower bound

LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2

1 minus 16γ K Tinfinθ4 H2κminus2H0

We now choose K = 2cM(γ Tinfinκ2) so that the second term vanishes The prefac-tor of the last term is then given by 32Mcθ4κ2 Choosing θ small enough so thatθ4 lt κ2(256c) say we finally obtain the lower bound

LW1 ge M K

16H2κminus1W1 gt 0

valid outside of some sufficiently large compact set as required 13


6 Convergence Speed Towards the Invariant Measure

In this section we are concerned with the convergence rates towards the invariant mea-sure in the case 1 lt k le 2 where it exists Our main result will be that k = 4

3 isthe threshold separating between exponential convergence and stretched exponentialconvergence

61 Upper bounds Our main tool for upper bounds will be the integrability boundsobtained in the previous section together with the results recently obtained in [DFG06BCG08]

The results obtained in Sect 5 suggest that it is natural to work in spaces of func-tions weighted by exp(δV ε) where V was defined in (418) For ε gt 0 and δ gt 0 wetherefore define the space B(ε δ) as the closure of the space of all smooth compactlysupported functions under the norm

ϕ(εδ) = supxisinR4|ϕ(x)| exp

(minusδH ε(x))

where we used the letter x to denote the coordinates (p0 q0 p1 q1) Note that the dualnorm on measures is a weighted total variation norm with weight exp(δH ε(x)) Wealso say that a Markov semigroup Pt with invariant measure micro has a spectral gap in aBanach space B containing constants if there exist constants C and γ such that

Ptϕ minus micro(ϕ)B le Ceminusγ tϕB forallϕ isin BAs a consequence of the bounds of Sect 5 we obtain

Theorem 61 Let k isin (1 2] and set κ = 2k minus 1 Then the semigroup Pt extends to a

C0-semigroup on the space B(ε δ) provided that ε le max 1

2 1minus κ Furthermore

a If 1 lt k lt 43 then for every ε isin [1minus κ κ) and every δ gt 0 the semigroup Pt has a

spectral gap in B(ε δ) Furthermore there exists δ0 gt 0 such that it has a spectralgap in B(κ δ) for every δ le δ0In particular for every δ gt 0 there exist constants C gt 0 and γ gt 0 such that thebound

Pt (x middot )minus microTV le C exp(δH1minusκ(x))eminusγ t (61)

holds uniformly over all initial conditions x and all times t ge 0b If k = 4

3 then there exists δ0 gt 0 such that the semigroup Pt has a spectral gap inB( 1

2 δ) for every δ le δ0 In particular there exists δ gt 0 such that the convergenceresult (61) holds

c For 43 lt k lt 2 there exist positive constants δ C and γ such that the bound

Pt (x middot )minus microTV le C exp(δHκ(x))eminusγ tκ(1minusκ) (62)

holds uniformly over all initial conditions x and all times t ge 0d For the case k = 2 set ζ as in Theorem 58 Then for every Tinfin lt α2C and everyζ lt ζ there exists C gt 0 such that the bound

Pt (x middot )minus microTV le C H1+ζ (x)tminusζ (63)

holds uniformly over all initial conditions x and all times t ge 0

168 M Hairer

Proof The set of bounded continuous functions is dense in B(ε δ) and is mapped intoitself by Pt Therefore in order to show that it extends to a C0-semigroup on B(ε δ) itremains to verify that

1 There exists a constant C such that Ptϕ(εδ) le Cϕ(εδ) for every t isin [0 1] andevery bounded continuous function ϕ

2 For every ϕ isin Cinfin0 one has limtrarr0 Ptϕ minus ϕ(εδ) = 0

Using the a priori bounds on the solutions given by the bound LH le γ (T + Tinfin) it ispossible to check that the second statement holds for every (ε δ) The first claim thenfollows from [MT93] and (518)

It remains to show claims a to d Claims a and b follow immediately from (518) Toshow that claim c also holds we use the fact that by using (518) in the case ε = 2

k minus 1we can find δ gt 0 such that the bound

LW le minusδ2V 2κminus1W = minusδ k2minusk W (log(W ))2minus

1κ

holds outside of some compact subset of R4 Since we are considering a regular Markovprocess every compact set is petite This shows that there exists a constant δ such thatin the terminology of [BCG08] W is a ϕ-Lyapunov function for our model with

ϕ(t) = δ k2minusk t (log t)2minus

1κ

In particular this yields the identity

Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s

1κminus2ds = C(log t)

1minusκκ

for some constant C depending on δ and κ It follows from the results in [BCG08] thatthe convergence rate to the invariant measure is given by

ψ(t) = 1

(ϕ Hminus1ϕ )(t)

= Ct1minus2κ1minusκ eminusγ tκ(1minusκ)

for some positive constants C and γ so that (62) followsThe case k = 2 can be treated in a very similar way It follows from the first part of

the proof of Theorem 58 that there exists β gt 0 and a function W growing like H1+ζ

at infinity such that one has the bound LW le minusβH ζ outside of some sufficiently large

compact set Therefore W is a ϕ-Lyapunov function for ϕ(t) = minusβtζ

1+ζ Following thesame calculations as before we obtainψ(t) = Ctminusζ so that the required bound followsat once 13

62 Lower bounds In order to be able to use Theorem 37 we need upper bounds on themoments of some observable that is not integrable with respect to the invariant measureThis is achieved by the following proposition

Proposition 62 For every α gt 0 and every κ isin [0 12 ] there exist constants Cα and Cκ

such that the bounds(Pt Hα

)(x) le (H(x) + Cαt)α

(Pt expαHκ)(x) le exp

(αHκ(x) + Cκ(1 + t)κ(1minusκ)

)

hold for every t gt 0 and every x isin R4


Proof Note first that LH le γ (T + Tinfin) and that

T (partp0 H)2 + Tinfin(partp1 H)2 = T p20 + Tinfin p2

1 le 2(T + Tinfin)H (64)

It follows that for α ge 1 there exists C gt 0 such that one has the bound

d

dt

(Pt Hα)(x) = (Pt (LHα)

)(x)

= α(Pt (H

αminus1LH + γ (α minus 1)Hαminus2(T p20 + Tinfin p2

1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α

The last inequality followed from the concavity of x rarr x1minus 1α Setting Cα = Cα the

bound on Pt Hα now follows from a simple differential inequality The correspondingbound for α isin (0 1) follows by a simple application of Jensenrsquos inequality

The bounds on the exponential of the energy are obtained in a similar way Set

fκ(x) = x(log x)2minus 1κ and note that there exists a constant Kκ such that provided that

κ isin (0 12 ] fκ is concave for x ge exp(αK κ

κ ) It then follows as before from (64) andthe bound on LH that there exists a constant C such that

d

dt

(Pt expα(Kκ + H)κ)(x) le C

(Pt (Kκ + H)2κminus1 expα(Kκ + H)κ

)(x)

= C(Pt fκ(expα(Kκ + H)κ)

)(x)

le C fκ((Pt expα(Kκ + H)κ

)(x)

) (65)

The result then follows again from a simple differential inequality 13As a consequence we have the following result in the case k = 2

Theorem 63 For every ζ gt ζ and every x0 isin R4 there exists a constant C and asequence tn increasing to infinity such that micro minus microtn ge Ctminusζn

Proof Let ζ isin (ζ ζ ) and let ε gt 0α gt ζ(1+ε) It then follows from Theorem 58 andProposition 62 that the assumptions of Theorem 37 are satisfied with W (x) = H ζ (x)h(s) = sminus1minusε F(s) = sαζ and g(x0 t) = (H(x0) + Ct)α Applying Theorem 37yields the lower bound

micro minus microtn ge Ctminus (1+ε)αζαminusζminusεζ

n

for some C gt 0 and some sequence tn increasing to infinity Choosing ε sufficientlysmall and α sufficiently large we can ensure that the exponent appearing in this expres-sion is larger than minusζ so that the claim follows 13

Furthermore we have

Theorem 64 Let k isin ( 43 2) and define κ = 2

k minus 1 Then there exists a constant c suchthat for every initial condition x0 isin R4 there exists a constant C and a sequence oftimes tn increasing to infinity such that micro minus microtn ge C exp(minusctκ(1minusκ)n )

170 M Hairer

Proof We apply Theorem 37 in a similar way to above but it turns out that we donrsquotneed to make such lsquosharprsquo choices for h and F Take h(s) = sminus2 F(s) = s3 andlet W = exp(K Hκ) with the constant K large enough so that W is not integrablewith respect to micro It then follows from Proposition 62 that we can choose g(x t) =exp

(3K Hκ(x) + C(1 + t)κ(1minusκ)

)for a suitable constant C The requested bound fol-

lows at once noting that h (F middot h) g = 1g2 13

7 The Case of a Weak Pinning Potential

In this section we are going to study the case k le 1 that is when we have eitherV1 asymp V2 or V1 V2 at infinity This case was studied extensively in the previous works[EH00RT02EH03Car07] but the results and techniques obtained there do not seemto cover the situation at hand where one of the heat baths is at lsquoinfinite temperaturersquoFurthermore these works do not cover the case k lt 12 where one does not have aspectral gap and exponential convergence fails One further interest of the present workis that unlike in the above-mentioned works we are able to work with the generatorL instead of having to obtain bounds on the semigroup Pt This makes the argumentsomewhat cleaner

We divide this part into two subsections We first treat the case where one can find aspectral gap which is relatively easy in the present setting In the second part we thentreat the case where the spectral gap fails to hold which follows more closely the heu-ristics set out in Sect 22 There we also show that rather unsurprisingly no invariantmeasure exists in the case where k le 0

71 The case k gt 12 Our aim is to find a modified version H of the energy functionH such that for a sufficiently small constant β0 one has exp(minusβ0 H)L exp(β0 H) 0at infinity This is achieved by the following result

Theorem 71 Let k isin ( 12 1) and let δ isin [ 1k minus1 1] Then there exist constants cC gt 0

β0 gt 0 and a function H R4 rarr R such that

bull The bounds cH le H le C H hold outside of some compact setbull For any t gt 0 the operator Pt admits a spectral gap in the space of measurable

functions weighted by exp(β0 H δ)

Remark 72 Combining this result with Proposition 51 shows the existence of constantscC gt 0 such that

intexp(cH) dmicro ltinfin but

intexp(C H) dmicro = infin

Remark 73 The technique used in the proof of Theorem 71 is more robust than thatused in the previous sections In particular it applies to chains of arbitrary length Itwould also not be too difficult to modify it to suit the more general class of potentialsconsidered in [RT02Car07]

Proof Define the variable y = (q p0 p1) with q = (q0 minus q1)2 and let A and B bethe matrices defined by

Adef=

⎛

⎝0 1

2 minus 12minus2α minusγ 0

2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)


With this notation we can write the equations of motion for y following from (12) as

dy = Ay dt + F(y Q) dt + B dw(t) (72)

where we defined the centre of mass Q = (q0 + q1)2 and F R4 rarr R3 is a vector-valued function whose components are all bounded by C + |V prime1(q0)| + |V prime1(q0)| for someconstant C

Since det A = minusγα lt 0 and we know from a simple contradiction argument [RT02Car07] that the energy of the system converges to zero under the deterministic equationy = Ay we conclude that all eigenvalues of A have strictly negative real part As a con-sequence there exists γ gt 0 such that the strictly positive definite symmetric quadraticform

〈y Sy〉 def=int infin

0eγ teAt y2 dt (73)

is well-defined A simple change of variable shows that one then has the bound

〈eAt SeAt y〉 le eminusγ t 〈y Sy〉 (74)

For any given (small) value ε gt 0 let now Gε Rrarr R be a smooth function such that

bull There exists a constant Cε such that the bounds Gε(q)V prime1(q) le Cε minus |V prime1(q)|2 and|Gε(q)|2 le Cε + |V prime1(q)|2 hold for every q isin R

bull One has |G primeε(q)| le ε for every q isin R

Since we assumed that k lt 1 it is possible to construct a function Gε satisfying theseconditions by choosing Rε sufficiently large setting Gε(q) = minusV prime1(q) for |q| ge 2RεGε(q) = q|Rε|2kminus2 for |q| le Rε and interpolating smoothly in between For largevalues of Rε one can then guarantee that |G primeε(q)| le C R2kminus2

ε which does indeed go to0 for large values of Rε

We now define for a (large) constant ξ to be determined

H = H + 〈y Sy〉 minus ξ(p0 + p1)(Gε(q0) + Gε(q1))

Before we bound LH we note that we have the bound

(Gε(q0) + Gε(q1))(V prime1(q0) + V prime1(q1)

)

= 2(Gε(q0)V

prime1(q0) + Gε(q1)V

prime1(q1)

)+

(int q1

q0

G primeε(q) dq

)(V prime1(q0)minus V prime1(q1)

)

le 2Cε minus 2(|V prime1(q0)|2 + |V prime1(q1)|2

)+ Cε(q0 minus q1)

2

for some constant C independent of εIt therefore follows from (74) (72) (12) and the properties of Gε that there exist

constants Ci independent of ξ and ε such that we have the bound

LH le C1 minus γ p20 minus γ 〈y Sy〉 + 2〈y SF(y Q)〉

+ξ(Gε(q0) + Gε(q1))(V prime1(q0) + V prime1(q1)

)

+γ ξp0(Gε(q0) + Gε(q1))minus ξ(p0 + p1)(G primeε(q0)p0 + G primeε(q1)p1

)

le C2

(Cε + |V prime1(q0)|2 + |V prime1(q1)|2

)

minus γ minus C3εξ

2〈y Sy〉 minus 2ξ

(|V prime1(q0)|2 + |V prime1(q1)|2

)

172 M Hairer

It follows that by first making ξ sufficiently large and then making ε sufficiently smallit is possible to obtain the bound

LH le C minus γ2

(1 + 〈y Sy〉 + |V prime1(q0)|2 + |V prime1(q1)|2

) (75)

for some constant C (The constant C depends of course on the choice of ξ and of ε butassume those to be fixed from now on) Furthermore it follows immediately from thedefinition of H that

(H H) le C(

1 + 〈y Sy〉 + |V prime1(q0)|2 + |V prime1(q1)|2)le C H2minus 1

k (76)

where we used the scaling behaviour of V1 in order to obtain the second bound Setnow W = exp(β0 H δ) for a constant β0 to be determined It follows from (517) that thebound

LW le β0δW H δminus1(LH + 2β0δ H δminus1(H H)

)

holds outside of some sufficiently large compact set Combining this with (76) and (75)we see that if δ isin [ 1k minus 1 1] and β0 is sufficiently small then the bound

LW le minusCW (H)δ+1minus 1k le minusCW

holds outside of some compact set The claim then follows immediately fromTheorem 34 13

The case k = 1 can be shown similarly but the result that we obtain is slightlystronger in the sense that one has a spectral gap in spaces weighted by H δ for any δ gt 0

Theorem 74 Let k = 1 and let δ gt 0 Then for any t gt 0 the operator Pt admits aspectral gap in the space of measurable functions weighted by H δ

Proof The proof is similar to the above but this time by setting y = (q0 q1 p0 p1)

Adef=

⎛

⎜⎝

0 0 1 00 0 0 1minusα α minusγ 0α minusα 0 0

⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

d y = A y dt + F(y) dt + B dw(t)

for some bounded function F It then suffices to construct S similarly to above and toset H = 〈y S y〉 without requiring any correction term This yields the existence ofconstants C1 and C2 such that one has the bounds

LH le minusC1 H (H H) le C2 H

outside of some compact set The existence of a spectral gap in spaces weighted by H δ

follows at once The claim then follows from the fact that H is bounded from above andfrom below by multiples of H 13


72 The case k le 12 This case is slightly more subtle since the function V prime(q) iseither bounded or even converges to zero at infinity so that bounds of the type (75) arenot very useful We nevertheless have the following result

Theorem 75 Let k isin (0 12 ] Then (12) admits a unique invariant probability measure

micro and there exist constants cC gt 0 β0 gt 0 and a function H R4 rarr R such that

bull The bounds cH le H le C H hold outside of some compact setbull If k = 1

2 then Pt admits a spectral gap in the space of measurable functions weighted

by exp(β0 H)bull If k lt 1

2 then there exist positive constants C and γ such that the bound

Pt (x middot )minus microTV le C exp(β0 H(x))eminusγ tk(1minusk) (77)


Proof Define again y A and B as in (71) but let us be slightly more careful aboutthe remainder term We define as before the center of mass Q = (q0 + q1)2 and thedisplacement q = (q0 minus q1)2 and write

V prime1(q0) = V prime1(Q) + R0(q Q) V prime1(q1) = V prime1(Q) + R1(q Q)

With this notation defining furthermore the vector 1 = (0 1 1) the equation of motionfor y = (q p0 p1) is given by

dy = Ay dt minus V prime1(Q)1 dt + R(Q y) dt + B dw(t) R = (0minusR0(Q y)minusR1(Q y))

This suggests the introduction for fixed Q isin R of the reduced generator LQ acting onfunctions from R3 to R by

LQ = 〈Ay party〉 minus V prime1(Q)〈1 party〉 +1

2〈Blowastparty Blowastparty〉

Following the usual procedure in the theory of homogenisation we wish to correctthe lsquoslow variablersquo Q in order to obtain an effective equation that takes into accountthe behaviour of the lsquofast variablersquo y Since the equation of motion for Q is given byQ = (p0 + p1)2 = 〈1 y〉2 this can be achieved by finding a function ψ(Q) suchthat 〈1 y〉2 minus ψ(Q) is centred with respect to the invariant measure for LQ and thensolving the Poisson equation LQϕQ = 〈1 y〉2minus ψ(Q)

Since all the coefficients of LQ are linear (remember that Q is a constant there) thiscan be solved explicitly yielding

ψ(Q) = minus 2

γV prime1(Q) ϕQ(y) = minus〈a y〉 a = (1 1γ 1γ )

We now introduce the corrected variable Q = Q +〈a y〉 so that the equations of motionfor Q are given by

d Q = minus 2

γV prime1(Q) dt + 〈a R(Q y)〉 dt +

radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ

the lsquomean temperaturersquo T = (T + Tinfin)2 and

R = 〈a R(Q y)〉 + γ(

V prime1(Q)minus V prime1(Q)) (78)

we thus see that there exists a Wiener process W such that Q satisfies the equation

d Q = minusγ V prime1(Q) dt + R dt +radic

2γ T dW (t)

Setting again S as in (73) this suggests that in order to extract the tail behaviour ofthe invariant measure for (12) a good test function would be V1(Q) + 〈y Sy〉 Thisfunction however turns out not to be suitable in the regime where Q is large and y issmall because of the constant appearing when applying L to 〈y Sy〉 In order to avoidthis let us introduce a smooth increasing function χ R+ rarr [0 1] such that χ(t) = 1

for t ge 2 and χ(t) = 0 for χ le 1 We also define the function 〈Q〉 =radic

1 + Q2 so that

|V prime1(Q)| le C〈Q〉2kminus1 and similarly for V primeprime1 (Q)Note that since we are considering the regime where V prime1 is a bounded function there

exists a constant CS such that

minusCS minus 2γ 〈y Sy〉 le L〈y Sy〉 le CS minus γ2〈y Sy〉

where γ is as in (74) Furthermore we note that since all terms contained in R are ofthe form V prime1(Q)minusV prime1(Q + 〈b y〉) for some vector b isin R3 there exists a constant C suchthat the bound

|R| le

C |Q| le C |y|C |y|〈Q〉2kminus2 |Q| ge C |y| (79)

holds for every pair (Q y) (In particular R is bounded) We now set

W = exp (β0〈y Sy〉) + exp(β0λV1(Q)

)

for some positive constants β0 and λ to be determinedSince we are only interested in bounds that hold outside of a compact set we use

in the remainder of this proof the notation f g to signify that there exists a constantc gt 0 such that the bound f le cg holds outside of a sufficiently large compact setWith this notation one can check in a straightforward way that there exist constants βidepending on β0 and λ such that the two-sided bound

exp(β1 H) W exp(β2 H)

holdsIt follows then from the chain rule that there exist constants Ci gt 0 such that one

has the upper bound

LW le β0

(CS minus ( γ

2minus C1β0)〈y Sy〉

)exp (β0〈y Sy〉)

+β0λ(minus(1minus C2β0λ)|V prime1(Q)|2 + C3|V prime1(Q)||R| + C4V primeprime1 (Q)

)exp

(β0λV1(Q)

)

(710)


Choosing β0 sufficiently small we obtain the existence of a constant C such that thebound

LW (

C minus 〈y Sy〉)

exp (β0〈y Sy〉)+〈Q〉2kminus1(

C |R| minus 〈Q〉2kminus1)

exp(β0λV1(Q)

)

holdsWe now consider three separate cases In the regime λV1(Q) ge 〈y Sy〉 ge C and

provided that λ is chosen sufficiently small it follows from (79) that we have the bound

LW 〈Q〉2kminus1(


exp(β0λV1(Q)

) minus〈Q〉4kminus2W

In the regime where λV1(Q) ge 〈y Sy〉 but 〈y Sy〉 le C we similarly have

LW C exp(β0C)minus 〈Q〉4kminus2 exp(β0λV1(Q)


Finally in the regime where λV1(Q) le 〈y Sy〉 we have the bound

LW minus〈y Sy〉 exp (β0〈y Sy〉) minus|y|2W

Combining all of these bounds we have

LW minus(log W )2minus1k W

so that the upper bounds on the transition probabilities follow just as in the proof ofTheorem 61 with κ replaced by k 13

Before we obtain lower bounds on the convergence speed we show the followingnon-integrability result

Lemma 76 In the case k lt 12 there exists β gt 0 such that

intexp(βV1(Q)) dmicro = infin

Proof We are going to construct functions W1 and W2 satisfying Wonhamrsquos criterionLet Q S and y be as in the proof of the previous result and set

W2 = exp(2βV1(Q)) + exp(ε〈y Sy〉)for constants β gt 0 and ε gt 0 to be determined It follows from the boundedness of V prime1V primeprime1 and R that whatever the choice of β one has

LW2 exp(βV1(Q))

provided that we choose ε sufficiently small Setting

W1 = exp(βV1(Q))minus exp(ε〈y Sy〉)we have similarly to (710) the bound

LW1 ge β((C2β minus 1)|V prime1(Q)|2 minus C3|V prime1(Q)||R| + C4V primeprime1 (Q)

)exp(βV1(Q))

+ε((γ 2minus C1ε)〈y Sy〉 minus CS) exp(ε〈y Sy〉)so that an analysis similar to before shows that LW1 ge 0 outside of some compact setprovided that ε lt γ (2C1) and β gt 1C2 thus concluding the proof 13Remark 77 The proof of Lemma 76 does not require k gt 0 It therefore shows thatthere exists no invariant probabilitymeasure for (12) if k le 0

176 M Hairer

We now use this result in order to obtain the following lower bound on the convergenceof the transition probabilities towards the invariant measure

Theorem 78 Let k isin (0 12 ) Then there exists a constant c such that for every initial

condition x0 isin R4 there exists a constant C and a sequence of times tn increasing toinfinity such that micro minus microtn ge C exp(minusctk(1minusk)

n )

Proof We use the same notations as above Let β be sufficiently large so that the func-tion exp(βV1(Q)) is not integrable with respect to the invariant measure We also fixsome small ε gt 0 and we set

W = exp(βV1(Q)) + exp(ε〈y Sy〉)We then obtain in a very similar way to before the upper bound

LW le β((C2β minus 1)|V prime1(Q)|2 + C3|V prime1(Q)||R| + C4V primeprime1 (Q)

)exp(βV1(Q))

+ ε(CS minus (γ 2minus C1ε)〈y Sy〉) exp(ε〈y Sy〉)It follows again from a similar analysis that there exists a constant C gt 0 such that thebound

LW le C(log W )2minus1k W

holds outside of some compact set As in the proof of Proposition 62 this implies theexistence of a constant C gt 0 such that one has the pointwise bound

Pt W le W exp(

C(1 + t)k(1minusk))

Combining this with Lemma 76 the rest of the proof is identical to that of Theorem 6413

Acknowledgements The author would like to thank Jean-Pierre Eckmann Xue-Mei Li Jonathan Matting-ly and Eric Vanden-Eijnden for stimulating discussions on this and closely related problems as well asCharles Manson for discovering several mistakes in an earlier version This work was supported by an EPSRCAdvanced Research Fellowship (grant number EPD0715931)

References

[BCG08] Bakry D Cattiaux P Guillin A Rate of convergence for ergodic continuous Markovprocesses Lyapunov versus Poincareacute J Funct Anal 254(3) 727ndash759 (2008)

[Bon69] Bony J-M Principe du maximum ineacutegalite de Harnack et uniciteacute du problegraveme de Cauchypour les opeacuterateurs elliptiques deacutegeacuteneacutereacutes Ann Inst Fourier (Grenoble) 19 no fasc 1277ndash304 xii (1969)

[Car07] Carmona P Existence and uniqueness of an invariant measure for a chain of oscillators incontact with two heat baths Stoch Process Appl 117(8) 1076ndash1092 (2007)

[CGGR08] Cattiaux P Gozlan N Guillin A Roberto C Functional inequalities for heavy tailsdistributions and application to isoperimetry httparxivorgabs08073112v1[mathPR]2008

[CGWW07] Cattiaux P Guillin A Wang F-Y Wu L Lyapunov conditions for logarithmic Sobolevand super Poincareacute inequality httparxivorgabs07120235[mathPR] 2007

[DFG06] Douc R Fort G Guillin A Subgeometric rates of convergence of f -ergodic strongMarkov processes httparxivorgabsmath0605791v1[mathST] 2006


[DMP+07] DeVille REL Milewski PA Pignol RJ Tabak EG Vanden-Eijnden E Nonequi-librium statistics of a reduced model for energy transfer in waves Comm Pure ApplMath 60(3) 439ndash461 (2007)

[DPZ96] Da Prato G Zabczyk J Ergodicity for Infinite-Dimensional Systems Vol 229 of LondonMathematical Society Lecture Note Series Cambridge Cambridge University Press 1996

[DV01] Desvillettes L Villani C On the trend to global equilibrium in spatially inhomogeneousentropy-dissipating systems the linear Fokker-Planck equation Comm Pure Appl Math54(1) 1ndash42 (2001)

[EH00] Eckmann J-P Hairer M Non-equilibrium statistical mechanics of strongly anharmonicchains of oscillators Commun Math Phys 212(1) 105ndash164 (2000)

[EH03] Eckmann J-P Hairer M Spectral properties of hypoelliptic operators Commun MathPhys 235(2) 233ndash253 (2003)

[EPR99a] Eckmann J-P Pillet C-A Rey-Bellet L Entropy production in nonlinear thermallydriven hamiltonian systems J Statist Phys 95(1-2) 305ndash331 (1999)

[EPR99b] Eckmann J-P Pillet C-A Rey-Bellet L Non-equilibrium statistical mechanics ofanharmonic chains coupled to two heat baths at different temperatures Commun MathPhys 201(3) 657ndash697 (1999)

[FR05] Fort G Roberts GO Subgeometric ergodicity of strong Markov processes Ann ApplProbab 15(2) 1565ndash1589 (2005)

[Hai05] Hairer M A probabilistic argument for the controllability of conservative systems httparxivorgabsmath-ph0506064v2 2005

[HM08a] Hairer M Mattingly J Slow energy dissipation in anharmonic oscillator chains httparxivorgabs07123889v2[math-ph] 2009

[HM08b] Hairer M Mattingly J Yet another look at Harrisrsquo ergodic theorem for Markov chainshttparxivorgabs08102777v1[mathPR] 2008

[HN04] Heacuterau F Nier F Isotropic hypoellipticity and trend to equilibrium for the fokker-planckequation with a high-degree potential Arch Rat Mech Anal 171(2) 151ndash218 (2004)

[HN05] Helffer B Nier F Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operatorsand Witten Laplacians Vol 1862 of Lecture Notes in Mathematics Berlin Springer-Verlag2005

[Houmlr67] Houmlrmander L Hypoelliptic second order differential equations Acta Math 119 147ndash171(1967)

[Houmlr85] Houmlrmander L The Analysis of Linear Partial Differential Operators III Vol 274 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemat-ical Sciences] Berlin Springer-Verlag 1985

[MA94] MacKay RS Aubry S Proof of existence of breathers for time-reversible or hamiltoniannetworks of weakly coupled oscillators Nonlinearity 7(6) 1623ndash1643 (1994)

[MT93] Meyn SP Tweedie RL Markov Chains and Stochastic Stability Communications andControl Engineering Series London Springer-Verlag London Ltd 1993

[MTVE02] Milewski PA Tabak EG Vanden-Eijnden E Resonant wave interaction with randomforcing and dissipation Stud Appl Math 108(1) 123ndash144 (2002)

[RT00] Rey-Bellet L Thomas LE Asymptotic behavior of thermal nonequilibrium steady statesfor a driven chain of anharmonic oscillators Commun Math Phys 215(1) 1ndash24 (2000)

[RT02] Rey-Bellet L Thomas LE Exponential convergence to non-equilibrium stationary statesin classical statistical mechanics Commun Math Phys 225(2) 305ndash329 (2002)

[RW01] Roumlckner M Wang F-Y Weak Poincareacute inequalities and L2-convergence rates of Markovsemigroups J Funct Anal 185(2) 564ndash603 (2001)

[Ver00] Veretennikov AY On polynomial mixing estimates for stochastic differential equationswith a gradient drift Teor Veroyatnost i Primenen 45(1) 163ndash166 (2000)

[Ver06] Veretennikov AY On lower bounds for mixing coefficients of Markov diffusions In FromStochastic Calculus to Mathematical Finance Berlin Springer 2006 pp 623ndash633

[Vil07] Villani C Hypocoercive diffusion operators Boll Unione Mat Ital Sez B Artic Ric Mat(8) 10(2) 257ndash275 (2007)

[Vil08] Villani C Hypocoercivity 2008 To appear in Memoirs Amer Math Soc[VK04] Veretennikov AY Klokov SA On the subexponential rate of mixing for Markov pro-

cesses Teor Veroyatn Primen 49(1) 21ndash35 (2004)[Won66] Wonham WM Liapunov criteria for weak stochastic stability J Diff Eqs 2 195ndash207

(1966)

Communicated by A Kupiainen


Abstract

Introduction

Heuristic Derivation of the Main Results

A Potpourri of Test Function Techniques

Existence and Non-existence of an Invariant Probability Measure

Integrability Properties of the Invariant Measure

Convergence Speed Towards the Invariant Measure

The Case of a Weak Pinning Potential

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132 M Hairer

potential V1 and interacting through a harmonic force that is the Hamiltonian systemwith Hamiltonian

H(p q) = p20 + p2

1

2+ V1(q0) + V1(q1) + V2(q0 minus q1) V2(q) = α

2q2 (11)

We assume that the first particle is in contact with a Langevin heat path at temperatureT gt 0 The second particle is also assumed to have a stochastic force acting on it butno corresponding friction term so that it is at lsquoinfinite temperaturersquo The correspondingequations of motion are

dqi = pi dt i = 0 1dp0 = minusV prime1(q0) dt + α(q1 minus q0) dt minus γ p0 dt +

radic2γ T dw0(t) (12)

dp1 = minusV prime1(q1) dt + α(q0 minus q1) dt +radic

2γ Tinfin dw1(t)

where w0 and w1 are two independent Wiener processes Although we use the symbolTinfin in the diffusion coefficient appearing in the second oscillator this should not beinterpreted as a physical temperature since the corresponding friction term is missingso that detailed balance does not hold even if Tinfin = T We also assume without furthermention throughout this article that the parameters α γ T and Tinfin appearing in themodel (12) are all strictly positive

The equations of motion (12) determine a diffusion on R4 with generator L givenby

L = X H minus γ p0partp0 + γ(

T0part2p0

+ Tinfinpart2p1

) (13)

where X H is the Liouville operator associated to H ie the first-order differential oper-ator corresponding to the Hamiltonian vector field It is easy to show that (12) has aunique global solution for every initial condition since the evolution of the total energyis controlled by

LH = γ (T0 + Tinfin)minus γ p20 (14)

and so EH(t) le H(0) exp (γ (T0 + Tinfin)t) Schematically the system under consider-ation can thus be depicted as follows where we show the three terms contributing to thechange of the total energy

This model is very closely related to the toy model of heat conduction previously stud-ied by various authors in [EPR99bEPR99aEH00RT00RT02EH03Car07HM08a]consisting of a chain of N anharmonic oscillators coupled at its endpoints to two heat







V1(x) = |x |2k

2k+ R1(x)


supxisinR[minus11]

supmle2


ltinfin




134 M Hairer















ζ = 3

4


κ = 2

kminus 1 (15)






)exp(δHκ )




1kminus1

)



)exp(δH)



bull One hasint









2nminus1


136 M Hairer

0 02 04 06 08 1 12

0

50

100 Energy

Time (times105)

0 02 04 06 08 1 12

0

1000

2000

3000Energy

Time (times105)

0 02 04 06 08 1 12

0

05

1


Time ( times105)





T +Tinfin minus 12



k for
















2 dW (t) X ge 1




138 M Hairer








2 dW (t) (21)





2 plusmnε Xη+1+ε


)exp


(δXσ+1

)



(δX1minusσ )















H1



q1(t) asymp H1

2k1 Q(H

12minus 1


12

1 P(H12minus 1

2k1 t) (22)


140 M Hairer



21 13p(H

12minus 1


32kminus1

1 13q(H12minus 1

2k1 t)

(23)



2γ T dw0(t) (24)


H0 = p20

2+ V1(q0) +

α

2q2

0



2kminus1

1 〈132p〉





2kminus1

1




of the type


p〉H2kminus1

1

)dt +





d X =(

2

Kminus 1

)1


p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)





η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction







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V1(x) = |x |2k

2k+ R1(x)


supxisinR[minus11]

supmle2


ltinfin




134 M Hairer















ζ = 3

4


κ = 2

kminus 1 (15)






)exp(δHκ )




1kminus1

)



)exp(δH)



bull One hasint









2nminus1


136 M Hairer

0 02 04 06 08 1 12

0

50

100 Energy

Time (times105)

0 02 04 06 08 1 12

0

1000

2000

3000Energy

Time (times105)

0 02 04 06 08 1 12

0

05

1


Time ( times105)





T +Tinfin minus 12



k for
















2 dW (t) X ge 1




138 M Hairer








2 dW (t) (21)





2 plusmnε Xη+1+ε


)exp


(δXσ+1

)



(δX1minusσ )















H1



q1(t) asymp H1

2k1 Q(H

12minus 1


12

1 P(H12minus 1

2k1 t) (22)


140 M Hairer



21 13p(H

12minus 1


32kminus1

1 13q(H12minus 1

2k1 t)

(23)



2γ T dw0(t) (24)


H0 = p20

2+ V1(q0) +

α

2q2

0



2kminus1

1 〈132p〉





2kminus1

1




of the type


p〉H2kminus1

1

)dt +





d X =(

2

Kminus 1

)1


p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)





η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








134 M Hairer















ζ = 3

4


κ = 2

kminus 1 (15)






)exp(δHκ )




1kminus1

)



)exp(δH)



bull One hasint









2nminus1


136 M Hairer

0 02 04 06 08 1 12

0

50

100 Energy

Time (times105)

0 02 04 06 08 1 12

0

1000

2000

3000Energy

Time (times105)

0 02 04 06 08 1 12

0

05

1


Time ( times105)





T +Tinfin minus 12



k for
















2 dW (t) X ge 1




138 M Hairer








2 dW (t) (21)





2 plusmnε Xη+1+ε


)exp


(δXσ+1

)



(δX1minusσ )















H1



q1(t) asymp H1

2k1 Q(H

12minus 1


12

1 P(H12minus 1

2k1 t) (22)


140 M Hairer



21 13p(H

12minus 1


32kminus1

1 13q(H12minus 1

2k1 t)

(23)



2γ T dw0(t) (24)


H0 = p20

2+ V1(q0) +

α

2q2

0



2kminus1

1 〈132p〉





2kminus1

1




of the type


p〉H2kminus1

1

)dt +





d X =(

2

Kminus 1

)1


p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)





η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction










ζ = 3

4


κ = 2

kminus 1 (15)






)exp(δHκ )




1kminus1

)



)exp(δH)



bull One hasint









2nminus1


136 M Hairer

0 02 04 06 08 1 12

0

50

100 Energy

Time (times105)

0 02 04 06 08 1 12

0

1000

2000

3000Energy

Time (times105)

0 02 04 06 08 1 12

0

05

1


Time ( times105)





T +Tinfin minus 12



k for
















2 dW (t) X ge 1




138 M Hairer








2 dW (t) (21)





2 plusmnε Xη+1+ε


)exp


(δXσ+1

)



(δX1minusσ )















H1



q1(t) asymp H1

2k1 Q(H

12minus 1


12

1 P(H12minus 1

2k1 t) (22)


140 M Hairer



21 13p(H

12minus 1


32kminus1

1 13q(H12minus 1

2k1 t)

(23)



2γ T dw0(t) (24)


H0 = p20

2+ V1(q0) +

α

2q2

0



2kminus1

1 〈132p〉





2kminus1

1




of the type


p〉H2kminus1

1

)dt +





d X =(

2

Kminus 1

)1


p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)





η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








136 M Hairer

0 02 04 06 08 1 12

0

50

100 Energy

Time (times105)

0 02 04 06 08 1 12

0

1000

2000

3000Energy

Time (times105)

0 02 04 06 08 1 12

0

05

1


Time ( times105)





T +Tinfin minus 12



k for
















2 dW (t) X ge 1




138 M Hairer








2 dW (t) (21)





2 plusmnε Xη+1+ε


)exp


(δXσ+1

)



(δX1minusσ )















H1



q1(t) asymp H1

2k1 Q(H

12minus 1


12

1 P(H12minus 1

2k1 t) (22)


140 M Hairer



21 13p(H

12minus 1


32kminus1

1 13q(H12minus 1

2k1 t)

(23)



2γ T dw0(t) (24)


H0 = p20

2+ V1(q0) +

α

2q2

0



2kminus1

1 〈132p〉





2kminus1

1




of the type


p〉H2kminus1

1

)dt +





d X =(

2

Kminus 1

)1


p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)





η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction




















2 dW (t) X ge 1




138 M Hairer








2 dW (t) (21)





2 plusmnε Xη+1+ε


)exp


(δXσ+1

)



(δX1minusσ )















H1



q1(t) asymp H1

2k1 Q(H

12minus 1


12

1 P(H12minus 1

2k1 t) (22)


140 M Hairer



21 13p(H

12minus 1


32kminus1

1 13q(H12minus 1

2k1 t)

(23)



2γ T dw0(t) (24)


H0 = p20

2+ V1(q0) +

α

2q2

0



2kminus1

1 〈132p〉





2kminus1

1




of the type


p〉H2kminus1

1

)dt +





d X =(

2

Kminus 1

)1


p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)





η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








138 M Hairer








2 dW (t) (21)





2 plusmnε Xη+1+ε


)exp


(δXσ+1

)



(δX1minusσ )















H1



q1(t) asymp H1

2k1 Q(H

12minus 1


12

1 P(H12minus 1

2k1 t) (22)


140 M Hairer



21 13p(H

12minus 1


32kminus1

1 13q(H12minus 1

2k1 t)

(23)



2γ T dw0(t) (24)


H0 = p20

2+ V1(q0) +

α

2q2

0



2kminus1

1 〈132p〉





2kminus1

1




of the type


p〉H2kminus1

1

)dt +





d X =(

2

Kminus 1

)1


p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)





η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction


















H1



q1(t) asymp H1

2k1 Q(H

12minus 1


12

1 P(H12minus 1

2k1 t) (22)


140 M Hairer



21 13p(H

12minus 1


32kminus1

1 13q(H12minus 1

2k1 t)

(23)



2γ T dw0(t) (24)


H0 = p20

2+ V1(q0) +

α

2q2

0



2kminus1

1 〈132p〉





2kminus1

1




of the type


p〉H2kminus1

1

)dt +





d X =(

2

Kminus 1

)1


p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)





η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








140 M Hairer



21 13p(H

12minus 1


32kminus1

1 13q(H12minus 1

2k1 t)

(23)



2γ T dw0(t) (24)


H0 = p20

2+ V1(q0) +

α

2q2

0



2kminus1

1 〈132p〉





2kminus1

1




of the type


p〉H2kminus1

1

)dt +





d X =(

2

Kminus 1

)1


p〉radicTinfinK

(γ TinfinK X2

4

) 2kminus 3

2

+radic

2 dw1(t) (27)





η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction










η = 1minus 2

K+

2α2〈132p〉

TinfinK= 3

2

α2〈132p〉

Tinfinminus 1

2






Q = q0 + q1

2 q = q1 minus q0

2








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








142 M Hairer




i=1



L = X0 +1

2

msum

i=1









dz

dt= f0(z(t)) +

msum

i=1

fi (z(t))ui (t) z(0) = x0














lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction




















lim supRrarrinfin

supH(x)=R W1(x)

infH(x)=R W2(x)= 0





144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








144 M Hairer


ϕV = supxisinRn

|ϕ(x)|1 + V (x)




LV le minuscV











int t1 (1ϕ(s)) ds









) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction
















) (32)




y











)


146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








146 M Hairer

Proof Sinceint



int infin


int infin

1h(w) dw ltinfin








limErarrinfin

1

Esup

H(pq)=EW1(p q) = 0 (41)














T p20 + Tinfin p2

1

) (44)









lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction
















lim supErarrinfin

1

Einf

H(x)=EH0(x) lt 1 (45)




0)





p〉 from Sect 21


Hf (P Q) = P2

2+|Q|2k

2k




148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








148 M Hairer


L0 = Q minusR(P Q)








dq0 = p0 dt + α dt


2γ T dw0(t) (47)


P dt



H0 = p20


q2

2 (48)





2

+θq0


P)




(θ +

θγ 2

4α+ ε

)p2








(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction












(θ +

θγ 2

4α+ ε

)p2





ψE ( p0 q0) = ψ(

Hf ( p0 q0)

E

) ψ primeE =

1

Eψ prime

(Hf

E

) ψ primeprimeE =

1

E2ψprimeprime(

Hf

E

)





f g f sim g (411)













150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








150 M Hairer







(0oraminus1)minus1










k or b = 12 minus 1

k and a lt 1

Proof We can write









d le 1k minus 1







2k

and d le 1k minus 1

2




P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction











P)partP f


)partP f

+γ(









2k





Hminus 1




L0 = 2 minus C + R(Hf (P Q)) (415)




)ψE ( p0 q0) (416)








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








152 M Hairer





(1 + α2(partP)





13






















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction


















f (p1 q1)

)holds





+θαq0


P)


)




αθdef= αγ

(α minus γ θ

4

) cθ


2θ)


154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








154 M Hairer





f (P Q) + R(P Q)


2kminus 32

f and


2kminus1f



(1 + Hf ( p0 q0)

(1 + Hf (p1 q1))η

) (420)



LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)+ cθ ( p0minus L( p013(p1 q1)ψ))


LV minusCθ

(Hf ( p0 q0) + H

2kminus1

f (p1 q1)

)




(421)




(E0

Eη1

)∣∣∣∣ le C


(E0

Eη1

)∣∣∣∣ le C E

minus η21


(E0

Eη1

)∣∣∣∣ le C E

minus 12

1




|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction











|partmx part



|LE0| le C

(E0 + E

1minus 12k

0 E1kminus 1

21

)

|LE1| le C

(E

12 + 1

2k1 + E

12k0 E

121

)




121



|L p0| le C

(E

1minus 12k

0 + E1kminus 1

21

)


|13(p1 q1)L p0| le C

(E

1minus 12k

0 E3

2kminus11 + E

52kminus 3

21

)




2kminus 32

1

(E

12k0 + E

12k1

)le C

(E

12k0 + E

2kminus 3

21

) (423)


21


| p0L13| le C E120 E

32kminus 3

21

(E

12k0 + E

1minus 12k

1

)



(E

12k + 1

2 + 1η( 3

2kminus 32 )

0 + E12 + 1

η( 1

kminus 12 )

0

)

156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








156 M Hairer




LV minusC

(E0 + E

2kminus11

)




13






LW2 = γβ2W2


0 + β2

(T p2

0 + Tinfin p21

))le exp(βH)





exp(βHf ( p0 q0)


β lt 1T










0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction












0 le H0 le (1 + ε)H + Cε (51)














P F)(p1 q1) +




p0 = p0 +p(p1 q1) q0 = q0 +q(p1 q1)ψ(E0Eη1 ) (54)



)dt


+radic


)dw1(t) +



2γ T dw0


)dt

+Kp dt +(


dt +radic



2


L0(1)p = αQ

(1minus ψ(

Hf (P Q)E)) (56)

158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








158 M Hairer












dq0 = p0 dt +radic


)dw1(t) +




2γ T dw0

+Rp dt + Kp dt +(


dt +radic


(57)


H0 = p20

2+ Veff (q0) + θ p0q0 + C0





E0 le C(

E0 + E3minus2k1

) E0 le C

(E0 + E3minus2k

1

)







)qpartp0ψ




2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction










2k0 E



32kminus1minus η





)partp1(qψ)


bounded by E120 E


2k1minusk




ηminus32 (1minus 1

k )




+γ Tinfin((partPp)



+γ T(


)


)+


)


) (59)



E0 + Eδ1

)holds where





(E

12k0 + E

32kminus11




∣∣∣ le C


)H

32kminus1

f (p1 q1)

le C

(1 + |q0|2kminus2 + E


1

)E

32kminus11

le C

(E

1minus 1k

0 E3

2kminus11 + E


1

)

160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








160 M Hairer


p0


∣∣∣ le C

(E0 E

η( 12minus 1

k )+3

2kminus11 + E

120 E


1

)





2k + 1η( 1

kminus 12 )





has the bound

(partp0 H0)2 le (1 + θ) p2

0 + Cθ + θ2H0


LW

βW= LH0 + γβ

(T (partp0 H0)



0 + Cθ2H0






ζ gt ζdef= 3

4










(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction














(partp0 V

)2 = (1minus c)2 p20 + R0



(partp0 V



(partp0 V


(partp1 V

)2 le p21 + εV


L0(P Q) = 3P2 minus 4Hf (510)




L0 = P2 minus 4

3Hf + R (511)



4

3Hf (p1 q1) + R(p1 q1) +


)(partP

)(p1 q1)


Hf (p1 q1) le (1 + ε)V

162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








162 M Hairer


H34



4

3(1 + ε)V







T(partp0 V


)2 minus TinfinL)


)



)+ ζγ (ζ + 1)V ζ

(T ε + Tinfinε +

4

3Tinfin(1 + ε)

)


le γ (ζ + 1)


4

3ζTinfin + K ε

)V ζ



supH(pq)=E







(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction










(partp1 V

)2 le p21 minus εV


LW0 ge γ (ζ + 1)


4


0)

)V ζ




so that

LW1 ge γ (ζ + 1)


4


)V ζ





intp2

1 micro(dx) = infin


(1 + Hf (p0 q0) + p2

1

)





R4exp

(δH

2kminus1(x)



164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








164 M Hairer





the bound

LW = δκW(



) (517)


|partp0 V | le C

(E

120 + E

120 E


32kminus11

)le C

(E

120 + E

121

)le Cradic

V

|partp1 V | le C

(E

121 + E

12k0 + E

52kminus21 + E

120 E

32kminus 3

21

)le C

(E

120 + E

121

)le Cradic

V




) (518)



int

R4exp

(H

2kminus1(x)





set W2(x) = exp(

K Hκ(x))


LW2



0


(T p2

0 + Tinfin p21

)

le C(

1 + H2κminus1)




L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction










L0 = Hf minus k + 1

2kP2 + R



W1 = exp(

K(


def= exp(K H1)


1 le H0 le (1 + ε)H



LW1

K W1= LH1 + γ K

(T (partp0 H1)







T p20 + Tinfin p2

1

)


T p20 + Tinfin p2

1

)





2minus κ2

)Hκminus1



LH1 Hκminus1(γ(κ

2minus 2

)p2

0 + 2(γ minus 2θ) p20 + 2θH0

)+ M H2κminus2L

166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








166 M Hairer


LH1 Hκminus1(


)+ M H2κminus2L0



LH1 Hκminus1(

2θH0 minus M

2Hκ

f (p1 q1)


1)

minusM

2Hκminus1 Hκ




LH1 M


1 (520)






(partp1 H1)2 ge κ2

2H2κminus2 p2



LW1

K W1 M

4H2κminus1 +

(γ K Tinfinκ2

2minus cM

)H2κminus2 p2



LW1 ge M K

16H2κminus1W1 gt 0









(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction















(minusδH ε(x))

















168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








168 M Hairer








1κ



1κ


Hϕ(t) =int t

1

ds

ϕ(s)= δminus k

2minusk

int log(t)

0s


1minusκκ


ψ(t) = 1

(ϕ Hminus1ϕ )(t)













)







d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction













d

dt


)(x)

= α(Pt (H


1)))(x)

le C(Pt Hαminus1

)(x) le C

((Pt Hα)(x)

)1minus 1α







d

dt



)(x)


)(x)


)(x)

) (65)





n


Furthermore we have



170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








170 M Hairer


















Adef=

⎛

⎝0 1


2α 0 0

⎞

⎠ Bdef= radic

2γ

⎛

⎝0 0radicT 0

0radic

Tinfin

⎞

⎠ (71)



















)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction


























)

= 2(Gε(q0)V


prime1(q1)

)+

(int q1

q0

G primeε(q) dq


)


)+ Cε(q0 minus q1)

2





)


)

le C2


)




)

172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








172 M Hairer


LH le C minus γ2


) (75)


(H H) le C(


k (76)



)







Adef=

⎛

⎜⎝


⎞

⎟⎠ B

def= radic2γ

⎛

⎜⎜⎝

0 00 0radicT 0

0radic

Tinfin

⎞

⎟⎟⎠

and noting that

























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction



























ψ(Q) = minus 2



d Q = minus 2


radic2T

γdw0(t) +

radic2Tinfinγ

dw1(t)

174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








174 M Hairer

Defining γ = 2γ


R = 〈a R(Q y)〉 + γ(




2γ T dW (t)



1 + Q2 so that





|R| le




)





has the upper bound

LW le β0

(CS minus ( γ




)exp

(β0λV1(Q)

)

(710)



LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction










LW (

C minus 〈y Sy〉)



exp(β0λV1(Q)

)



LW 〈Q〉2kminus1(


exp(β0λV1(Q)















LW2 exp(βV1(Q))




)exp(βV1(Q))


176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction








176 M Hairer




n )




)exp(βV1(Q))




Pt W le W exp(

C(1 + t)k(1minusk))



References


































(1966)



Abstract

Introduction



































(1966)



Abstract

Introduction








how hot can a heat bath get?

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