on - institut für analysis und scientific computingarnold/papers/sobolev_alt.pdf · toscani dimat...
TRANSCRIPT
On logarithmic Sobolev inequalities�
Csisz�ar�Kullback inequalities� and the rate of
convergence to equilibrium for Fokker�Planck
type equations
June �� ����
Anton Arnold�� Peter Markowich�� Giuseppe Toscani��Andreas Unterreiter�
�Fachbereich Mathematik� TU�Berlin� MA ���� D����� Berlin� Germany
�Institut fur Analysis und Numerik� Universitat Linz� A��� Linz� Austria
�Dipartimento di Matematica� Universit�a di Pavia� I���� Pavia� Italy
�Fachbereich Mathematik� Universitat Kaiserslautern� D���� � Kaiserslautern�Germany
arnold�math�tu�berlin�de� markowich�numa�uni�linz�ac�at�toscani�dimat�unipv�it� unterreiter�mathematik�uni�kl�de
�
Acknowledgements
This research was partially supported by the grants ERBFMRXCT��� � �TMR�Network� from the EU� the bilateral DAAD�Vigoni Program� the DFG underGrant�No� MA ��������� and the NSF �DMS�� � ��� The �rst author ack�nowledges fruitful discussions with L� Gross and D� Stroock� the second authorwith D� Bakry� and the second and third authors interactions with C� Villani�
AMS ���� Subject Classi�cation� � K �� � B�� ��D�� ��D�� �J�
Abstract
It is well known that the analysis of the large�time asymptotics of
Fokker�Planck type equations by the entropy method is closely related to
proving the validity of logarithmic Sobolev inequalities� Here we highlight
this connection from an applied PDE point of view�
In our uni�ed presentation of the theory we present new results to thefollowing topics� an elementary derivation of Bakry�Emery type condi�
tions� results concerning perturbations of invariant measures with gene�
ral admissible entropies� sharpness of convex Sobolev inequalities� explicit
construction of optimal Csisz�ar�Kullback inequalities� applications to non�
symmetric linear �e�g� kinetic� and certain non�linear Fokker�Planck type
equations �Desai�Zwanzig model� drift�diusion�Poisson model�
Contents
� Introduction �
� Decay of the relative Entropy as t�� �
��� Spectral gap of the symmetrized equation � � � � � � � � � � � � � �
��� Admissible relative entropies and their generatingfunctions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��
��� Exponential decay of the entropy production and the relative entropy ��
��� Convex Sobolev inequalities in entropy version � � � � � � � � � � � ��
�� Non�symmetric Fokker�Planck equations � � � � � � � � � � � � � � �
�
� Sobolev Inequalities ��
��� Three versions of convex Sobolev inequalities � � � � � � � � � � � � ��
��� Perturbation lemmata for the potential A�x� � � � � � � � � � � � � �
��� Poincar�e�type inequalities � � � � � � � � � � � � � � � � � � � � � � ��
��� Sharpness results � � � � � � � � � � � � � � � � � � � � � � � � � � � �
� A generalized Csisz�ar�Kullback Inequality ��
��� Optimality � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
��� An extension of the inequality � � � � � � � � � � � � � � � � � � � � �
��� Comparison with the classical Csisz�ar�Kullback inequality � � � � �
��� An example � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
�� Asymptotic behavior of U as e��u�� � � � � � � � � � � � � � � �
��� Passing from weak to strong convergence in L��d�� � � � � � � � � �
��� The optimal inequality for the logarithmic entropy � � � � � � � � � �
��� Entropy�type estimates on kf � gkL��d�� � � � � � � � � � � � � � � �
� Nonlinear Model Problems �
�� Desai�Zwanzig type models � � � � � � � � � � � � � � � � � � � � � � ��
�� The drift�di�usion�Poisson model � � � � � � � � � � � � � � � � � � ��
Appendix Proofs of Section � ��
� Introduction
One of the fundamental problems in kinetic theory is the analysis of the timedecay �rate� of solutions of kinetic models towards their equilibrium states� Themaybe most often applied methodology in time asymptotics is the entropy me�thod� where the convergence towards equilibrium is concluded using the timemonotonicity of the physical entropy of the system� A classical example for thisapproach is provided by the spatially homogeneous Boltzmann equation� whereconvergence to the Maxwellian equilibrium state has been proven in this way �cf��e�g�� �CeIlPu���� Theorem ����� and the references therein��
�
To illustrate the entropy approach we shall now present two simple �even expli�citly solvable� kinetic model problems outlining the ideas which we will developin this paper�
At �rst we consider the Bhatnagar�Gross�Krook �BGK� model of gas dynamics�which is a �simpli�ed� version of the Boltzmann equation carrying to some extentthe same amount of physical information �BhGrKr ��� The IVP for the spacehomogeneous version reads�
�f
�t� J�f� �� �
�Mf � f� � v � IRn� t � � ����a�
f�v� t � � � fI�v�� v � IRn ����b�
�of course� only the case n � � is of physical interest�� with the normalizationRIRnfIdv � �� Here M
f �Mf �v� denotes the Maxwellian distribution function
Mf �v� � m ������n�� exp��jv � uj
�
��
������
with mass
m �
ZIRn
fdv�
mean velocity
u �
ZIRn
vfdv�m
and temperature
� �
ZIRn
�v � u��f�nm�
The relaxation rate � is assumed to be a positive constant� It is then an easyexercise to show that m� u and � are left invariant under the temporal evolutionof ������ such that these quantities can be computed from the initial data fI � andMf�t� �MfI follows�
The exponential L��IRnv ��convergence of f��� t� to its equilibrium state MfI with
rate � can be easily veri�ed by explicitly solving ������
For carrying out the entropy approach we consider the physical entropy �Boltz�mann�s H�functional�
H�f� �
ZIRn
f ln fdv �����
�
and the entropy production
I�f� �
ZIRn
ln fJ�f�dv� �����
with the easily veri�able relation
d
dtH�f�t�� � I�f�t��� ��� �
Since lnMf is quadratic in v and due to the conservation of mass� mean velocityand temperature� we calculate
I�f�t�� � �
ZIRn
ln f�t��Mf�t� � f�t�� dv
� ��ZIRn
�ln f�t�� lnMf�t�
� �f�t��Mf�t�
�dv
� ��ZIRn
ln
�f�t�
Mf�t�
�f�t�dv � �
ZIRn
ln
�Mf�t�
f�t�
�Mf�t�dv�
Since f�t� and Mf�t� have the same mass� Jensen�s inequality implies that bothterms are nonpositive� Thus we obtain the stronger version of Boltzmann�s H�Theorem
�I�f�t�� � �e�f�t�jMf�t�� �����
where the so called relative �to the Maxwellian� entropy is de�ned by
e�f jMf� ��
ZIRn
f ln
�f
Mf
�dv� �����
Note that e�f jMf� � i� f � Mf �Gibb�s Lemma�� Again� since lnMf isquadratic in v and since f�t� and Mf�t� have the same mass� velocity and tem�perature� we have
e�f�t�jMf�t�� � H�f�t���H�Mf�t���
and since Mf�t� �MfI we conclude from ��� �� �����
d
dte�f�t�jMf�t�
��d
dtH�f�t�� � I�f�t�� � ��e �f�t�jMf�t�
��
Exponential convergence to zero of the relative entropy with rate � follows im�mediately �for initial data which have �nite relative entropy�
e�f�t�jMfI � � e��te�fI jMfI �� t � � �����
The Csisz�ar�Kullback inequality �Csi�����Kul ��
kf �Mfk�L��IRn� � �e�f jMf � �����
then gives L��IRn��convergence to equilibrium �at the suboptimal rate ����
Summing up� we used the lower bound ����� of the �negative� entropy productionin terms of the relative entropy to explicitly control the convergence of the �rela�tive and absolute� entropies and to control the strong convergence of the solutionto its equilibrium state� The second model problem� which is a two�velocity ra�diative transfer model� demonstrates that the approach of the �rst example maynot be general enough� We consider the ODE in IR��
d�
dt�
� �� �� ��
��� t � ����a�
��� �
�uIvI
�����b�
where ��t� �
�u�t�v�t�
�and � Since we consider ����� as a kinetic model�
we assume � for physical reasons only � uI� vI � which implies u�t�� v�t� � �Obviously� u�t� and v�t� tend exponentially with rate � to their equilibriumstate w� � �uI vI���� To apply the entropy approach let � � ��s� be anysmooth strictly convex function� de�ned on IR�� with ���� � � We introducethe relative entropy generated by ��
e���j��� �����u
w�� ��
v
w��
�w� ������
where we denoted the steady state
�� ��w�w�
��
Di�erentiating ������ with respect to t and using ����� gives the entropy equation
d
dte����t�j��� � I����t�j��� ������
with the production term
I���j��� � ��u� v�����u
w��� ��� v
w��
�� ������
Since � is strictly convex we have I���j��� � with equality i� � � ���Clearly� now we cannot bound the entropy production ������ from below directly
�
by a multiple of the relative entropy ������� Therefore we consider the timeevolution of the entropy production� Di�erentiating ������ gives
d
dtI����t�j��� � R����t�j��� ������
with the entropy production rate
R����t�j��� � ��I����t�j��� � �u�t�� v�t���
w�
�����u�t�
w�� ����
v�t�
w��
��
���� �
We can now bound the entropy production rate from below by a multiple of theentropy production�
R����t�j��� � ��I����t�j���
and conclude exponential convergence with rate � of the entropy productionfrom ������
jI���t�j���j � jI��I j���je���t� ������
Inserting I����t�j��� from ���� � into ������ and using ������ gives after integra�tion from s � t to s �� �using limt�� e���t�j��� � limt�� I����t�j��� � �
e����t�j��� Z �
t
�u�s�� v�s����w�
�����u�s�
w�� ����
v�s�
w��
�ds � �I���t�j���
��
������
First of all� ������ gives the exponential decay with rate � of the relative entropye�� and hence again the suboptimal decay rate of ��t� towards ��� Moreover������� furnishes an inequality involving the strictly convex function �� i�e� weobtain the Sobolev�type inequality�
��uIw�
� ��vIw�
�
�w� � �
��uI � vI�
����uIw�
�� ��� vIw�
�
�������
for all uI� vI � by setting t � in ������� Also� ������ implies that equality in������ holds i� u�t� � v�t� for all t � which is equivalent to uI � vI � w��Clearly� the inequality ������ can easily be obtained in a direct way �cp� Example��� of �Gro����� however� the presented approach allows generalization to muchmore complex kinetic problems �as will be the subject of the next sections��Particularly� the second example clearly shows that in some cases the entropyequality itself is not su!cient to derive exponential decay of the relative entropy�and equivalently� Sobolev�type inequalities�� It may be necessary to consider
�
the evolution of the entropy production and to �nd a lower bound of the entropyproduction rate in terms of the entropy production�
This leads directly to the main subject of this paper� In the following sectionswe shall consider the IVP for Fokker�Planck type equations �cf� ������ and applythe methodology presented above�
More speci�cally� Section � will be concerned with the decay of the relativeentropy as t��� There we shall obtain conditions on the entropy� the di�u�sion matrix and on the velocity �eld which allows for exponential convergenceunder weak conditions on the initial data �bounded relative entropy�� Section �then is concerned with the derivation of various forms of Sobolev�type inequalitiesand with the analysis of conditions which guarantee that the inequalities �satu�rate� nontrivially� In Section � we generalize the Csisz�ar�Kullback inequality torather general convex entropies� and in Section we apply the developed theoryto nonlinear Fokker�Planck models�
We conclude this introduction with a brief �and incomplete� discussion of therelevant literature and of those issues where our approach di�ers from alreadyexisting ones� Most importantly� our approach is to some extent based on thework by D� Bakry and M� Emery �cf� e�g�� �BaEm���� �Bak���� �Bak����� whichprovides a very general framework for hypercontractive semigroups and generali�zed Sobolev�type inequalities using the so�called iterated gradient �"�� formalism�BaEm���� In fact� some of our results are concretizations of the Bakry�Emery cri�terion �cf� the references cited above� to Fokker�Planck type equations� However�the Bakry�Emery approach does not allow an explicit control of the remainderterm in the Sobolev inequalities� Let us explain this point in more detail� Care�ful reading of the radiative transfer�type example presented above shows that theanalysis of the time decay of the entropy production gives at the same time the#sharp$ decay of the relative entropy as well as the remainder in ������� Theknowledge of this remainder allows to identify in ������ the �unique� state sa�turating the Sobolev�type inequality� Hence� the entropy approach gives at thesame time a proof of a #convex$ inequality and all cases of equality�
As far as the classical Gross logarithmic Sobolev inequality is concerned �Gro� ��the identi�cation of the cases of equality is due to Carlen �Car���� His approachis based on a slightly di�erent method� which relies on information�type ine�qualities� and ultimately requires a deep investigation of properties of Gaussianfunctions� In the same paper� the connection between Gross� logarithmic Sobolevinequality and the linear Fokker�Planck equation� through the Fisher measure ofinformation and the Blachman�Stam inequality �Bla� �� �Sta �� has been fruit�fully developed� The entropy�entropy production approach for the same Fokker�Planck equation has been recently addressed in �Tos��b�� �Tos��a�� There� thecases of equality follow easily from the identi�cation of the remainder� Physicallyspeaking� the entropy approach put in evidence that the remainder in this type
�
of inequalities depends on the complete dynamics in time of the solution of the�mass�conserving� linear problem that generates the inequality itself� In otherwords� many Sobolev type inequalities can be interpreted as an estimate for therelative entropy of the initial state with respect to the steady state of a linearmass conserving system�
An excellent background reference for logarithmic Sobolev inequalities is theoverview paper �Gro���� General linear homogeneous radiative transfer equa�tions generating certain inequalities involving convex functions are analyzed in�GaMaUn���� A calculation of the logarithmic Sobolev constant for non�symmetricODEs in IR� �generalizing our second example above� is presented in the appen�dix of �DiSC���� Entropy production arguments for the decay of the solution ofthe heat equation in IRn towards the fundamental solution were used in �Tos��a��A further application in kinetic theory is to be found in the asymptotic analysisof the spatially homogeneous Boltzmann equation� which in the so called �gra�zing collisions limit� leads to the Landau�Fokker�Planck equation �Vil���� �Vil�����Tos��b�� There it is hoped that the decay results of the relative entropy mightgive an indication of the rate of decay towards equilibrium of the spatially homo�geneous Boltzmann equation�
� Decay of the relative Entropy as t��
We now consider the IVP for the Fokker�Planck type equation
�t � div�D�r� �rA��� x � IRn� t � ����a�
��t � � � �I � L��IRn�� ����b�
with A � A�x� su!ciently regular �i�e� A � W ���loc �IR
n� and e�A � L��IRn��We assume that the symmetric di�usion matrix D � D�x� � �dij�x�� is locallyuniformly positive de�nite on IRn and dij � W ���
loc �IRn�� i� j � �� � � � � n� Obviously
we have the conservation propertyZIRn
��x� t�dx �
ZIRn
�I�x�dx� �����
In a kinetic context� the independent variable x in ����� stands for the velocity�
In this Section we assume �without restriction of generality�ZIRn
�I�x�dx � ��
One easily sees that ����a� has the steady state
���x� � e�A�x� � L���IRn�� �����
�
assuming �w�r�o�g�� A to be normalized asRIRne�A�x�dx � �� We remark that �by
a simple minimum principle� �I�x� � implies ��x� t� � for all x � IRn� t � �In this Section we shall investigate the convergence of ��t� towards the steadystate �� in various norms and in relative entropy� In particular we are concernedwith equations ����a� that exhibit an exponential decay to the steady state ������
��� Spectral gap of the symmetrized equation
We transform equation ����� to symmetric form �on L��IRn��� Therefore we set
z �� ��p���
which satis�es the IVP
zt � div�Drz�� V �x�z� x � IRn� t �z�t � � � zI �� �I�
p�� on IRn� �����
Here� V �x� denotes the potential
V �x� � ����Tr�D
��A
�x��� �
��rA��DrA �divD� � rA�� x � IRn� t �
��� �
��A�x�
is the Hessian of A�x� and the superscript #�$ denotes transposition� Nowde�ne the Hamiltonian
Hz � �div�Drz� V z �����
on the domain
DQ �� fz � L��IRn�jQ�z� z� �g
of the quadratic form Q�z�� z�� �� �Hz�� z��L��IRn� given by
Q�z�� z�� ��
ZIRn
r��z�p��
�D�x�r
�z�p��
����dx�
�obtained from ����� by a simple calculation�� For the following we shall assumethat the di�usion matrix D�x� and the potential A�x� are such that H generatesa C��semigroup on L��IRn� �for various su!cient conditions see �ReSi����� Notethat H satis�es
Hz �p��N
�zp��
�� z � DQ�
�
where N is the Dirichlet form�type operator on L��IRn� d��� de�ned by
�Nu� v�L��IRn�d��� ��
ZIRn
r�uDrv ���dx�
�see �Gro����� For future reference we also remark that
�ZIRn
����L�� dx � Q
���p�����p��
��
��p�����p��
� DQ� �����
Here L denotes the appropriate extension of the Fokker�Planck type operator
L� �� div �D �r� �rA�� �Obviously the spectrum ��H� is contained in ����� Since Q�z� � i� z �const
p�� we conclude that the ground state z� � exp��A��� of H is non�
degenerate and that �� is the unique normalized steady state of ����a��
The solution of ����� can be written as
z�t� �p��
Z�����
e��td�P��Ip���� �����
where P� is the projection valued spectral measure of H� Of course� we assumehere �I�
p�� � L��IRn�� From this spectral representation we immediately con�
clude the convergence of z�t� top�� in L��IRn� as t��� Let us now consider
a HamiltonianH with a positive spectral gap � �i�e� distance of ��H�nfg fromthe eigenvalue �� For the case D�x� � I� the identity matrix� a simple su!cientcondition for a positive spectral gap is given by V � L�loc�IRn% IR�� V �x� boundedbelow and V �x��� for jxj� � �see e�g� �ReSi��a� Th� XIII������ In this casewe obtain exponential convergence�
kz�t��p��kL��IRn� � kzI �p��kL��IRn�e�t�� � �����
which implies exponential convergence of ��t� to �� in L��IRn��ZIRnj��t�� ��jdx �
ZIRn
p��j��t�� ��jp
��dx
� k��k��
L��IRn�kz�t��p��kL��IRn� � O�e
�t���� �����
This very simple approach of course only holds under the restrictive assumption�I�
p�� � L��IRn�� In order to weaken this restriction and to better illustrate the
convergence of ��t� towards �� we now investigate the decay in relative entropye���t�j��� de�ned by ������ In fact� we shall not only consider this logarithmic#physical entropy$� but a wider class of relative entropies that essentially lie�between� this physical entropy and the quadratic functional k��t�k�
L��IRn����� �dx���
��
For future reference we also consider a second symmetrization of ����� �on L��IRn� d����which is more commonly used in the probabilistic literature on this subject� Weset
� �� �����
which satis�es the IVP
�t � ���� div�D��r��� x � IRn� t �
�I �� �I��� � L��IRn� d���� ������
In terms of this symmetrized problem we shall extend the allowed initial data �Ifrom L��IRn� ��� to the Orlicz space L� logL�IR
n� ����
��� Admissible relative entropies and their generating
functions
De nition ���� Let J be either IR or IR�� Let � � C�J� C��J� �where Jdenotes the closure of J� satisfy the conditions
���� � � �����a�
��� � � ��� � on J� �����b�
������� � �
�����IV on J� �����c�
Let �� � L��IRn�� �� � L���IRn� withR��dx �
R��dx � � and ����� � J ���dx��
a�e� Then
e����j��� ��ZIRn������� ���dx� ������
is called an admissible relative entropy �of �� with respect to ��� with generatingfunction ��
Note that
e����j��� � ������
follows from Jensen�s inequality�
��
Remark ���� If � satis�es the conditions �����a������c� so does its �norma�
lization� e���� � ���� � ������� � �� and they both generate the same relativeentropy� e
e����j��� � e����j���� In the sequel we will therefore assume that thegenerator � of e� be normalized as
����� � � �����d�
This implies that the relative entropies e� and their generators � are bijectivelyrelated� Due to the convexity of � we have
� � on J� ���� �
We now list some typical examples of admissible relative entropies on J � IR��The physical relative entropy ����� is generated by ���� � � ln� � � � ratherthan by ���� � � ln�� It is a special case of
���� � ��� �� ln� �
� �� ��� � ��� � � � � � �����a�
For � p �
�p��� � ���� ��p � �� ��p � p�� ��p���� � ��� � � � � � � �����b�
and for p � �
���� � ��� � ���� � � �����c�
generate admissible relative entropies� In the last example we clearly have e���j��� ���k��k�L��IRn����� �dx��
� ���In the above de�nition we excluded linear entropy functionals as they would bezero due to the assumed normalizations of �� and ���
In the following Lemmata we derive important properties of admissible entropies�
Lemma ���� a� Admissible entropies are generated by strictly convex functions�� b� For J � IR all admissible entropies are generated by ���� c��
Proof� a� Let g��� �� ������
� � � J with g��� � ����� Then� condition�����c� is equivalent to
g����� � ������
whenever ������ � Since ������ excludes �positive� poles of g on J � we conclude��� on J �
b� g�� � and g on IR implies g��� � const � and the assertion follows�
��
Hence� our class of generating functions � coincides with those considered in�BaEm��� �up to the normalizations �����a�� �����d���
For the following we shall assume J � IR� and �I � � Except being reasonablefrom a kinetic viewpoint� this case allows for a richer mathematical structuresince only quadratic admissible entropies exist for J � IR�
The above examples �����a�� �����c� of admissible entropies include the two limi�ting cases for the asymptotic behavior �as � ��� of the generating function ��An admissible relative entropy e����j��� can be bounded below by a logarithmicsubentropy e� and bounded above by a quadratic superentropy e�
Lemma ���� Let � generate an admissible relative entropy with J � IR�� Thenthere exists a logarithmic�type function � ���� a� and a quadratic function ����� c� such that
���� � ���� � ����� � � J� �����a�
and hence
� e����j��� � e����j��� � e���j���� �����b�
� and � both satisfy ������ and thus generate respectively an admissible sub�and superentropy for e��
Proof� Since J � IR�� the function g from the proof of Lemma ��� satis�es
g � g� � � g�� � on J� ������
Now denote the derivatives of the given function � by
���� � � ����� � � ������ �� �� � ������� �� �� � � �����
From ������ we readily get the estimate
����� � � ����� � � �
�� g��� � �� �� � �
with � �� ������� � � � �� ��� ������� � � Integrating the corresponding
estimate for ��� � �g�
��� ���� � ������ ������� � ���� � �
������
we obtain with ����� the upper bound for ��
���� ������ ln� � � ��� � �
����� � ���� � �
�� ���� � ��� �� ����� � � ������
��
To derive the lower bound of � one integrates ������ twice to show ���� � �����For � the function ���� is given by �����a� with � � �
�� � �
��
If � � we set
���� ������ � ���� ������
The well�known Csisz�ar�Kullback inequality ��Csi���� �Kul ��� shows that thelogarithmic relative entropy ����� is a �measure� for the distance between twonormalized L���IR
n��functions ��� �� withRIRn��dx �
RIRn��dx � ��
�
�k�� � ��k�L��IRn� � e���j���� ������
In a simple calculation using the subentropy e� and Lemma ��� this inequalitycan be extended to any admissible entropy e�� For � � and � hence givenby �����a� we introduce the normalized function e� �� ���� ��
�� � � and estimate
using �������
�
�k�� � ��k�L��IRn� �
�������
ke�� ��k�L��IRn� �������e�e�j��� � �
��e����j���� ���� �
With �����b� this gives
�
�k�� � ��k�L��IRn� �
�
��e����j���� ������
with the notation �� � ������ � ������� If � � � ������ is easily derived with
the estimate ����� and with ������� A detailed analysis of Csisz�ar�Kullbackinequalities for a larger class of generating functions � will be the topic of x�� Inparticular we shall derive a sharper variant of �������
In our subsequent analysis we shall need the continuity of the relative entropy�
Lemma ���� Let the sequence �j � � �as j ��� in L���IRn� ���� �dx�� with the
normalizationR�jdx �
R�dx �
R��dx � �� Then for all admissible entropies
� with J � IR��
e���jj���� e���j��� as j��� ������
Proof� Since � � C�� �� there exists for any � a positive � � ���� such that
j������ �����j � for � ���� � ����� ������
�
Integrating ������ we readily obtain the estimate�
�C���� such that j�����j � C������ �� for � � �� ������
We shall �rst derive estimates on j��a�� ��b�j% a� b � J � for three di�erent casesof a and b� For a� b � � we use the mean value theorem and the monotonicity of�� to estimate�
j��a�� ��b�j � ja� bj�j���a�j j���b�j� � ja� bjC������ a b�� �����
For the next case assume a � �� b � �or vice versa�� With ����� this yields
j��a�� ��b�j � j��a�� ����j j����� ��b�j� �a� ��C������ a �� �� � b�C����
� ja� bj �C������ a b �� C����� � ������
with C���� �� supc��� �j�� ����c�j
�c ��Finally� for a� b � we have j��a�� ��b�j � from �������Using these three estimates we can now control the di�erence of relative entropies�For arbitrarily small � we estimate�
je���jj���� e���j���j�
Zf �j��
� or ���
� g
�� �j���� �
��
��
� ��dx ������
Zf �j��
� ���
� g
�� �j���� �
��
��
� ��dx� �C������ �� C�����
ZIRn
j�j � �jdx ������
C����
ZIRn
j�j � �jp��
�j �p��dx �
ZIRn
��dx
� �C������ �� C����� k�j � �kL��IRn�
C����k�j � �kL��IRn����� �dx��k�j �kL��IRn����� �dx�� �� ������
As j�� this last term converges to �� since L��IRn� ���� �dx���convergence impliesL��IRn��convergence� This �nishes the proof�
For future reference we state the following elementary result for generators ofadmissible relative entropies�
��
Lemma ��� The generator � of every admissible relative entropy e� satis�es�
a�
����
� ��
��� �
�
�� ����� � �����
� ��
��� �
�
�� � � ���� �
with the notation �� � ������
b�
���� � �����
��
��
��
��
��
��� �
��� � ��� � � �� � ������
���� � ������
�� ��
��
��� �
��� � ��� �� � � � ������
Proof� a� For the right inequality of ���� � we have to show that G���� ���� ���� � �� � ��� � � But this follows from G���� � G
����� � � G
������ �
������ � � � �see �������For the left inequality of ���� � we have to show that G���� �� � ���� � ������ � � Like before we have G���� � G
����� � and it remains to show
G������ � ���� � ����� � � � � ������
Since � generates an admissible entropy we have ������� � � �IV ��� � �see������� �����c��� Thus there exists a unique �� � ���� such that ������� on�� ��� and �
������ � on �������In the case � � ������ we have G������ � ������� �In the case � � �� ��� we rewrite �����c� as
� � �����
����
��and integrate over the interval ��� ��� � ���
� � � � � �������������
�������������
� � �������������
�
Letting tend �� we obtain G��� � ���� � ����� � � and this �nishes the proofof ���� ��
b� Applying the Gronwall lemma to the right inequality of ���� � �with �xed ���gives
���� � �������
��
��
��
��
��� �
��� � ���� �
�
�� � � �� �
��
and ������ follows from �� ��
For the proof of ������ we use the left inequality of ���� � and estimate�
����� � �����
��
��� �
�
�� �����
��
��
�� �
��
�� � � ������
Applying the Gronwall lemma to ������ gives
���� � ����� ��� ���� ln
�
�� �� �
���� � � ���
and ������ follows with the estimate lnx � x� ��
��� Exponential decay of the entropy production and the
relative entropy
With our notion of superentropies we can now show the convergence of ��t� to�� in relative entropy�
Lemma ���� Let e� be an admissible relative entropy and assumezI � �I�
p�� � L��IRn�� Then e����t�j���� as t���
Proof� With the notation �� � ������ we estimate using �����b�� �������
� e����t�j��� � e���t�j��� �ZIRn
���t���
����dx� � ��kz�t��p��k�L��IRn��
�����
and the assertion follows from ������
If the Hamiltonian H in ����� has a spectral gap � � then the exponentialdecay from ����� of course carries over to the relative entropy� In the above lemma�the assumption �I�
p�� � L��IRn� is unnaturally restrictive� The subsequent
analysis aims at considering initial data which only have �nite relative entropyand at proving explicit decay bounds for the relative entropy in terms of theinitial relative entropy� The �price� for this extension will be a condition on theevolution problem ����a� �see �A�� below� that is stronger than assuming H tohave a spectral gap�
We now proceed similarly to �Tos��b� and to example � of the introduction�Consider the entropy production
I����t�j��� �� ddte����t�j��� ������
��
and the entropy production rate
R����t�j��� �� ddtI����t�j���� ������
To facilitate the computations we rewrite ����a� in the following form�
�t � div�Du��� ������
with the notation u � r� ����� Di�erentiating the relative entropy e����t�j���
gives
I����t�j��� �ZIRn�� ���
��t dx� ������
By using ������ we obtain after an integration by parts
I����t�j��� � �ZIRn
��� ���
�u�Du�� dx � � ���� �
due to the positivity of D� Using ������ we compute �������
R����t�j��� � �ZIRn
���� ���
�div�Du���u
�Du dx
� �
ZIRn
��� ���
�u�Dut��dx� ������
Clearly� the computations which lead to ���� � and ������ are formal� However�they can easily be justi�ed for initial data �I � L��IRn% ���� �dx�� and for entropygenerators without singularity at � � by taking into account the semigroupproperty of the HamiltonianH and the partial integration formula ������ Generaladmissible entropies can easily be dealt with by a local cut�o� at � � �
We now return to proving the exponential decay of e����t�j��� under additionalassumptions on A and D� At �rst we shall derive an exponential decay rate forthe entropy production I� by using the special form of the entropy productionrate �������
At �rst we consider the case of a scalar di�usion� i�e� D�x� � ID�x��
Lemma ���� Let the initial condition �I � L��IRn% ���� �dx�� satisfy jI���I j���j � for the admissible entropy e�� Assume that the scalar coe�cients A�x� andD�x� of ����a� satisfy the condition
�A�� �� such that��� n�
� �DrD �rD �
��&D �rD � rA�I
D��A
�x� rA�rD rD �rA
�� �
�D
�x�� �I
��
�in the sense of positive de�nite matrices� x � IRn� Then the entropy productionconverges to � exponentially�
jI����t�j���j � e����tjI���I j���j� t � ������
Proof� After an integration by parts �which can be justi�ed as mentioned above�the �rst term of the entropy production rate ������ reads
R� �
ZIRn
�IV�eA��D�juj�e�Adx
�
ZIRn
�����eA��D�e�Au��u
�xudx
ZIRn
�����eA��e�ADjuj�u � rDdx�
We set
ut � r�eAdiv�De�Au��in the second term of ������� which becomes
R� � ��ZIRn����eA��De�A�u � r�Ddivu� u�r� �rD �rAD�u u��u
�x�rD �rAD��dx�
We di�erentiate u � r�Ddivu� � �u � rD�divu Du � r�divu�� and we expressDu � r�divu� according to
�
�&�Djuj�� � �
�juj�&D �rD��u
�xu D
Xi�j
��ui�xj
��
Du � r�divu��
We rewrite R� as R� � S� S� with
S� � �ZIRn����eA��div�Dr�Djuj��e�A�dx
�
ZIRn
�����eA���D�u � r�juj�� DrD � ujuj�� e�Adx�
S� � �
ZIRn
����eA��De�A�u�r� �rAD �rD�u
�
�&Djuj� � �
�juj�rD � rA �
D
�� n��u � rD��
dx
�
ZIRn
����eA��e�A�n� ���u � rD�� �D�u � rD�divu
�DrD��u�xu D�
Xi�j
��ui�xj
��
�
�juj�jrDj�
dx
� T� T��
�
The second integral in S� can be written as
T� � �
ZIRn
����eA��e�AXi�j
�D�ui�xj
�
�
�D
�xiuj
�
�ui�D
�xj� ���ijrD � u
��
dx
and �A�� allows to estimate the �rst term in S� by
T� � ��ZIRn����eA��De�Ajuj�dx�
All in all we have
R� R� � R� S� S� � �R� S� T�� T�
�ZIRn
��IV�eA��D�juj� �����eA����D�u�
�u
�xu �Djuj�u � rD�
�����eA��Xi�j
�D�ui�xj
�
�
�D
�xiuj
�
�ui�D
�xj� ���ijrD � u
�� e�Adx
��
ZIRn
����eA��De�Ajuj�dx�
The �rst integral can be written asZIRn
tr�XY �e�Adx�
where X and Y are the �� ��matrices
X �
������eA�� ������eA��������eA�� �IV�eA��
�and� resp��
Y �
�� D�u� �u
�xu �
�Djuj�u � rD
D�u� �u�xu �
�Djuj�u � rD D�juj�
��
with
� �Xi�j
�D�ui�xj
�
�
�D
�xiuj
�
�ui�D
�xj� ���ijrD � u
��
�
X is non�negative de�nite since � generates an admissible entropy �cf� De�nition����� A simple calculation shows that Y is also nonnegative de�nite� Thus�Z
IRntr�XY �e�Adx �
��
and we have for the entropy production rate �������
R����t�j��� � ��ZIRn
����eA��De�Ajuj�dx � ���I����t�j����
The assertion now follows from
d
dtjI����t�j���j � ���jI����t�j���j� ������
In a special case of equation ����a� the condition �A�� has a simple geometricinterpretation�
Remark ���� For D�x� � I condition �A�� simply requires the uniform conve�xity of A�x� on IRn i�e�
�A�� �� such that��A�x��xi�xj
�i�j����� �n
� �I x � IRn�
The condition �A�� is a special case of the well�known Bakry�Emery condition forlogarithmic Sobolev�inequalities �BaEm���� �Bak���� �Bak���� In fact� the proof ofLemma ��� is a concretization of the approach of Bakry and Emery and was givenhere mainly for the sake of clarity� For general �symmetric and uniformly positivede�nite� di�usion matrices D�x� an �excursion� into basic di�erential geometryis� however� in order to understand the Bakry�Emery condition� Therefore weconsider the Riemannian manifold corresponding to the di�usion matrix D�x� ��dij�x�� as metric tensor� The Christo�el symbols are de�ned as the elements ofthe ��tensor�
"lij �nX
k�
�
�dkl��dij�xi
�dki�xj
� �dij�xk
�� ������
where dkl�x� are the entries of the inverse D�x��� � �dkl�x��� The Riemanncurvature tensor then reads
Rkilj �
�
�xi"ljk �
�
�xj"lik
nXm�
"lim"mjk �
nXm�
"ljm"mik ��� �
and the Ricci�tensor
�ij �nX
k�
Rikkj � ��� ��
��
The covariant derivative of a vector �eld X � �X�� � � � � Xn� is given by
riXj ��Xj
�xi
nXk�
"jikXk� ��� ��
We de�ne the symmetric ��tensor
�rSX�ij��
�
nXl�
�djlriXl dilrjX
l�� ��� ��
The Ricci tensor of the Fokker�Planck operator is de�ned as
Ricij�x� �nX
k�l�
dikdjl��kl �
�rSX�kl�x��
��� �a�
where we set
X i�x� � �nXj�
dij�
�xj
�A�x�� �
�ln detD�x�
�� ��� �b�
Then the Bakry�Emery condition for a general symmetric positive de�nite di�u�sion matrix reads
�A�� �� such that D�x�Ric�x� � �I x � IRn�
For the case D�x� � I we shall �rst compare �A��� or rather �A��� to the corre�sponding condition given in Prop� ��� of �Bak���� If D � I� their condition
�
mX�x��X�x� � m� n
m�Ric�x�� �D�x����
simpli�es to
�
mrA�rA � m� n
m
���A
�x�� �I
� x � IRn� ��� �
with some � � IR and m � �n��� denoting� respectively� the curvature anddimension of the FokkerPlanck operator L� If we assume � and m to be constant��� � only admits a global solution A�x�� x � IRn if m � �� Hence� ��� �reduces to �A�� with � � ��
We have
Lemma ����� If A � A�x��D � D�x� satisfy �A�� then the estimate ������holds�
��
Proof� See �BaEm���� �Bak����
Remark ����� To better understand the Bakry�Emery condition �A�� we consi�der a transformation of the Fokker�Planck type equation under an x � y di�eo�morphism
y � y�x� �� x � x�y�
on IRn� We denote J�y� � det �x�y�y� assume J on IRn and set
���y� t� � J�y���x�y�� t��
A lengthy calculation �cf� also �Ris���� gives
��t � divy�eD�ry�� ry� 'A� lnJ����� ��� �a�
where
eD�y� � �y�x�x�y��D�x�y��
��y
�x�x�y��
��� ��� �b�
'A�y� � A�x�y��� ��� �c�
Assume now that the Riemann curvature tensor ������ vanishes� Then the geo�metry produced by D is Euclidean and there exists a transformation y � y�x�
such that eD�y� � I �Ris���� The condition �A�� applied to ����a� reduces to �A��applied to ���� a� i�e� we need to assume the uniform convexity of 'A � lnJ inthe y�coordinates�
In one dimension �n � �� the Riemann curvature tensor always vanishes and thetransformation which yields a Fokker�Planck equation with di�usion coe�cient �can be constructed explicitly� It is de�ned by
dy
dx�x� �
pD�x�
��� � x � IR� ��� ��
For arbitrary n � an analogous transformation works if D is a constant matrix�We set
�y
�x�x� �
pD���� x � IRn� ��� ��
Then the transformed equation has the identity matrix as di�usion matrix�
In general �i�e� for a non�vanishing Riemann curvature tensor� the metric tensorcannot be transformed to the identity by a coordinate transformation� It can betransformed to the form eD�y� � I eD�y� where eD is a scalar function if isothermalcoordinates exist on the manifold corresponding to D� Locally at least this isalways the case for n � � �cf� �BeGo��� p������
��
From the exponential decay of the entropy production I� �Lemma ���� we shallnow derive the exponential decay of the relative entropy e��
Theorem ����� Let e� be an admissible relative entropy and assume thate���I j��� �� Let the coe�cients A�x� and D�x� satisfy condition �A��� Thenthe relative entropy converges to � exponentially�
e����t�j��� � e����te���I j���� t � ��� ��
Proof� We proceed in two steps and �rst derive ��� �� for �I � S �� f� �L���IR
n� ���� �dx�� I���j��� �g�
From the Lemmata ���� ���� we then know that e����t�j���� and I����t�j���� as t ��� Hence� integrating ������ �which also holds under condition �A�� �see �BaEm���� �Bak���� over �t��� gives
I��t� �d
dte��t� � ���e��t�� �����
which proves the assertion for su!ciently regular initial data�
For the general case we use a density argument to approximate �I in two steps� ��is given in L���IR
n�� and �I is a normalized �i�e�R�Idx �
R��dx � �� L���IR
n��
function with �nite relative entropy� �I � f� � L���IRn�e���j��� �g�
We �rst approximate �I by �N � L���IRn� withR�Ndx � ��N � IN��
�N�x� �� �N �I�x��lf�I����Ng
with the monotonously decreasing normalization constants �N �
� Rf�I����Ng
�Idx
��� � for N��� By construction we have j �N
��j � N�N � which implies �N �
L��IRn� ���� �dx���
�N converges to �I a�e�� and also in L��IRn��
k�I � �NkL��IRn� � j�� �N jZf�I����Ng
�Idx
Zf�I����Ng
�IdxN���� �
Now we construct a uniform bound for �� �N������ N � IN � we consider the convex�
monotonously increasing function �R���� � de�ned by �R��� � ��R� for � � R and �R��� � ���� for R �� with R � su!ciently large such that�R � � on IR�� This implies
�R��� � ���� ��R�� � � ������
�
From Lemma ���b we easily conclude with �� ��
�R���� � ��R��� ����� � � � ������
Since �N � � we have �N � ��I �for large N� and we estimate using ��������������
���N����� � �R��N
����� ������
� �R���I����� � ��R� �I
����� ����I � ��� �I
����� ���R��� ����I �� �����
Our assumptions on �I showRIRn����d� �� and Lebesgue�s dominated con�
vergence theorem gives
limN��
e���N j��� � e���I j���� ������
In the second approximation step we shall approximate �N in L���IR
n� ���� �dx��by the normalized �i�e�
R�N�Mdx � �� sequence f�N�MgM�IN � having �nite
entropy production� We choose �N�M � C����IRn� which denotes non�negativeC��functions with compact support in IRn�
We shall now show that
C����IRn� � S � fp� � L���IRn� ���� �dx��
jI���j���j �g� ���� �
We consider � withp� � C����IRn� and compact support () � supp� � IRn� Since
A � L�loc�IRn�� so is ��p�� � eA�� and hence ��
p�� � L��IRn� follows�
Using
������ � ���� ���� � �
which follows from ������� we estimate the entropy production �������
jI���j���j �Z�
�����
���uTD�x�u��dx
� ��kDkL����
Z�
�� ����jr� �
���j���dx
� ���kDkL����
Z�
�� ���
rr �
��
� dx� ���kDkL����
Z�
�� �
�����jrp�j� �jrAj��dx� ������
��
which is �nite since D�x�� ���� �x� and rA�x� � L�loc�IRn�� Hencep� � S�
Since C�� �IRn� is dense in L��IRn� ���� �dx��� �N can indeed be approximated by
f�N�Mg � S in L��IRn� ���� �dx��� Lemma �� then shows
limM��
e���N�M j��� � e���N j���� ������
From these two approximations we extract a normalized �diagonal� sequencef�N�M�N�g � S with
�N�M�N�N���� �I in L
��IRn�� �����a�
limN��
e���N�M�N�j��� � e���I j���� �����b�
For the approximations �N�M�N� we can apply ��� ���
e���N�M�N��t�j��� � e����te���N�M�N�j���� t � ������
where �N�M�N��t� denotes the solution of ����a� with initial data �N�M�N��
From the entropy bound ������ and from the Dunford�Pettis theorem �Ger��� weeasily conclude
�N�M�N��t�
��N���� ��t�
��in L��IRn� ���dx�� weakly�
We now use the lower semi�continuity of e���j��� to �nish the proof�ZIRn�
���t�
��
����dx� � lim inf
N��
ZIRn
�
��N�M�N��t�
��
����dx�
� e����te���I j���� t �
Due to the above density argument we do not have to make the #algebra hy�pothesis$ of �BaEm��� and x� of �Bak���� there one assumes that there exists acore for L that is stable under the evolution eLt and under composition with C�
functions� Using di�erent techniques such a density argument was also given inx� of �DeSt����
��
��� Convex Sobolev inequalities in entropy version
����� is the entropy version of a convex Sobolev inequalityZIRn
�
��
��
����dx� � �
��
ZIRn
�����
��
�r�
��
��
�Dr
��
��
����dx�
����a�
� � L���IRn� with
ZIRn
�dx �
ZIRn
��dx� ����b�
This inequality� of course� does not require our usual normalizationR��x�dx � ��
The relation of ����� to other versions of this inequality will be discussed in x��Note that L���IR
n� in ����b� can be replaced by L��IRn� if � is quadratic�
The desired L��convergence of ��t� to �� is now a direct consequence of Theorem���� and the Csisz�ar�Kullback inequality ������� �������
Corollary ����� Let e� be an admissible relative entropy and assume thate���I j��� �� Let the coe�cients A�x� and D�x� satisfy condition �A��� Thenthe solution of ����� satis�es
k��t�� ��kL��IRn� � e���tr�
��e���I j���� t � ������
with the notation �� � �������
To complete the line of argumentation we shall now show that the HamiltonianHhas a spectral gap if A andD satisfy �A��� This condition implies the logarithmicSobolev inequalities ����� for any admissible relative entropy e�� We start byrewriting the Sobolev�inequality ����� for ���� � � ln� � � � by settingr
�
���
jgjkgkL��IRn����dx��
�cf� also x��� We obtain after a simple calculationZIRn
jgj� ln jgj���dx� � �
�
ZIRn
rg�Drg���dx� kgk�L��IRn����dx�� ln kgkL��IRn����dx��
������
for all g � L��IRn� ���dx��� The well�known Rothaus�Simon mass gap theorem��Rot���� �Sim���� �Gro���� givesZ
IRn
rg�Drg���dx� � �kgk�L��IRn����dx�� �����a�
��
if ZIRn
g���dx� � � �����b�
Simple calculations show that
H ��p��Lp��� � DQ � L��IRn� dx�� L��IRn� dx�
where H is given by ����� andZIRn
rg�Drg���dx� �ZIRn
p��gH�
p��g�dx�
Using the spectral theorem it is easy to conclude that the spectral gap � of Hsatis�es � � �� i�e��
��H� �� �� � ��In many cases the logarithmic Sobolev constant � is indeed smaller than thespectral gap � �see� e�g�� example ������ x� of �DiSC���� and the discussions in�Rot���� �Bak���� �DeSt����
Remark ����� Assume that D�x� is pointwise in IRn bounded below by a sym�metric positive de�nite matrix D��x� i�e�
D��x� � D�x�� x � IRn�in the sense of positive�de�nite matrices� and that the Fokker�Planck operator
L���� �� div�D��r� �rA��satis�es the Bakry�Emery condition �A�� �with D replaced by D��� Then theconvex Sobolev inequality ������ holds with D replaced by D�� Since ��� � wealso have Z
IRn�����
��
�r�
��
��
�D�r
��
��
����dx�
�ZIRn
�����
��
�r�
��
��
�Dr
��
��
����dx� ������
and ������ follows� Consequently the decay statement in Lemma ���� Theorem���� and Corollary ���� and the above mass gap argument hold for L�
Note that this settles the case
D�x�I � D�x�� x � IRn�where D�x� and A�x� satisfy �A��� In particular for a uniformly positive de�nitedi�usion matrix
dI � D�x�� x � IRn�with d � IR�� and a uniformly convex potential A ��A�� holds� we obtain theSobolev inequality ������ where ������ has to be replaced by ����d���
��
��� Non�symmetric Fokker�Planck equations
At the end of this Section we extend the above analysis to the following class ofFokker�Planck type equations with certain rotational contributions to the drift�coe!cient� We consider
�t � div�D�r� ��rA �F ���� x � IRn� t � ���� �
��t � � � �I � ������
with the above conditions on �I � D and A� Additionally we assume for �F ��F �x� t� su!cient regularity and
div�D�F��� � on IRn � ����� ������
such that �� � e�A is still a stationary state of ���� �� Our objective is again toanalyze the rate of convergence of ��t� towards ��� For the symmetric problem����� we proceeded in x��� by deriving a di�erential inequality for the entropyproduction �see �������� In contrast� we shall here only apply the convex Sobolevinequality ����� that was obtained for the corresponding symmetric problem �i�e�
���� � with �F � ��
With the transformation � � ���� �cp� ������� ���� � may be rewritten as
�t � ���� div�D��r�� �F�Dr��
where the second term of the r�h�s� is skew�symmetric in L��IRn� d����
We �rst calculate�
d
dte����t�j��� � I����t�j��� T�
with I� as in ���� � and
T �
ZIRn
����
��
�div�D�F��dx�
We use ������ in the form
div�D�F �� � ��D�F �r�� ���
to obtain
T �
ZIRn
����
��
��D�F �r
��
��
���dx �
ZIRn
r����
��
�D�F��dx�
�
An integration by parts �nally gives
T � �ZIRn
�
��
��
�div�D�F���dx � �
By the convex Sobolev inequality ����� we obtain
d
dte����t�j��� � ���e����t�j���
and
e����t�j��� � e����te���I j���
follows again�
As an example of ���� � we consider the kinetic Fokker�Planck�type equation forthe distribution function f�x� v� t��
ft fA� fg � div�x�v��e�A�x�v�rx�v�eA�x�v�f��� t � �����a�
f�t � � � fI � L���IR�n�� �����b�
with the position variable x � IRn and the velocity variable v � IRn� Here
fA� fg �� rvA � rxf �rxA � rvf
denotes the Poisson bracket� For the sake of simplicity we set D � I� With thechoice �F � �rvA �rxA�
� the convergence of f�t� to its steady state f��x� v� �e�A�x�v� follows from the above calculations�
Theorem ����� Let fI � L���IR�n�� A � W ���loc �IR
�n� and � �RIR�n fI�x� v�dxdv �R
IR�n e�A�x�v�dxdv� Let e� be an admissible relative entropy and assume thate��fI jf�� �� Let A�x� v� be strictly convex i�e�
�� such that��A
��x� v��� �I x� v � IRn�
Then the relative entropy converges to � exponentially�
e��f�t�jf�� � e����te��fI jf��� t � ������
��
� Sobolev Inequalities
��� Three versions of convex Sobolev inequalities
We shall now discuss in detail the convex Sobolev inequality ������ In particularwe shall rewrite ����� in various equivalent forms and discuss its relation to otherknown inequalities� At �rst we set u � ����� Then ����� becomesZ
IRn
��u����dx� � �
��
ZIRn
����u�r�uDru���dx� ����a�
for all u � L���IRn� �L��IRn� if � is quadratic� which satisfyZIRnu���dx� �
ZIRn
���dx�� ����b�
Assume for the followingRIRn���dx� � ��
Hence setting u � v�RIRnv���dx�� we obtainZ
IRn
�
�vR
IRnv���dx�
����dx� � �
��
ZIRn
����
vRIRnv���dx�
� r�vDrv�RIRnv���dx���
���dx�
�����
for all v � L���d��� �� L���IRn� ���dx�� �v � L��d��� if � is quadratic��The most common form of convex Sobolev inequalities is obtained by settingv � f � in ������ This gives the so called steady state measure version�ZIRn
�
�f �
kfk�L��d���
����dx� � �
�
ZIRn
f �
kfk�L��d���
����
f �
kfk�L��d���
�r�fDrf ���dx�
�����
for all f � L��d����The inequalities ����������� hold for all functions �� � e�A �with L��dx��normequal ��� �� on IR
n and symmetric positive de�nite matrices D � D�x��which are su!ciently smooth �cf� Section �� and which satisfy �A�� �if D�x� �D�x�I�� or �A�� �if D�x� � I�� or �A��� or there exists a symmetric positivede�nite matrix D� � D��x� � D�x� such that A and D� satisfy �A���
If D�x� � I� ���x� �Ma�x� ���
���a�n��exp�� jxj�
�a� for some a then �A�� holds
with � � ��a and we obtain the celebrated Gross logarithmic Sobolev inequality�Gro� �� Z
IRn
f � ln
�f �
kfk�L��dMa�
�Ma�dx� � �a
ZIRn
jrf j�Ma�dx� �����
��
for all f � L��dMa�� where we set ���� � � ln� � � � �see also ��������A second example is provided by the same choices of �� and D setting ���� ��� � ���� We then haveZ
IRn
v�Ma�dx���Z
IRn
vMa�dx�
��
� aZIRn
jrvj�Ma�dx� ��� �
for all v � L��dMa�� This is an old inequality� and as remarked by Beckner �Bec���in one dimension was probably known to both mathematicians and physicists inthe ����s �Wey���� It has been a useful tool in di�erent subjects� like partialdi�erential equations �Nas �� and statistics �Che����
Finally let � � �p��� � �p � � � p�� � ��� see �����b�� Then we obtain the
inequalityZIRn
vpMa�dx���Z
IRnvMa�dx�
�p
� �ap� �p
ZIRn
jr�vp���j�Ma�dx� �����
for all v � L���dMa�� � p � �v � L��dMa� for p � ��� Setting juj � vp�� givesthe generalized Poincar�e�type inequality by Beckner �Bec����
p
p� ��Z
IRn
u�Ma�dx���Z
IRn
juj��pMa�dx�
�p�� �a
ZIRn
jruj�Ma�dx� �����
for all u � L��p�dMa�� � p � ��We remark that the inequalities ����� interpolate in a very sharp way between thePoincar�e�type inequality ��� � and the logarithmic Sobolev inequality ������ whichis obtained from ����� in the p� � limit� ����� and ����� represent a hierarchy ofconvex Sobolev inequalities� with the logarithmic Sobolev inequality ����� beingthe �strongest�� This� however� cannot be seen directly from ������ ����� andrequires a more involved line of argument �see �Bak���� p� �� where the spectralgap inequality ��� � is compared to the logarithmic Sobolev inequality ������� In�Led��� this interpolation is discussed for the Ornstein�Uhlenbeck process on IRn
and for the heat semigroup on spheres�
Note that the inequalities ������ ��� �� ������ ����� also hold when the Gaussianmeasure Ma is replaced by a general steady state �� � e�A� such that �A�D�satis�es the Bakry�Emery condition �A��� The quadratic form on the r�h�s� isthen replaced by �
��
R ru�Dru���dx��Let us now discuss brie*y some consequences of inequalities ������ ��� �� ������������
In spite of the fact that the class of admissible entropies and steady state measureswhich generate inequalities ������ ��� �� ������ ����� is very wide� Sobolev inequa�lities with respect to the Lebesgue measure follow only if ���� � � ln� � � ��
��
Actually� in this case the choice g� � f �Ma leads to the inequalityZIRn
g� ln
�g�
kgk�L��dx�
�dx
n n
�ln ��a
�kgk�L��dx� � �a
ZIRn
jrgj� dx �����
for all g � L��IRn�� a �cf� �Car����� Inequality ����� is the logarithmic Sobo�lev inequality for the heat kernel H� � �&� In Davies� book �Dav��� inequality����� is contained in the more restrictive Sobolev inequalities framework of ultra�contractive operators �see Theorem ����� and the rest of the Chapter�� To obtainDavies� form� we rewrite ����� as a family of logarithmic Sobolev inequalities for� �Z
IRn
g� ln g dx � �ZIRn
jrgj� dx MG���kgk�L��dx� kgk�L��dx� ln kgkL��dx� �����
for all � g � L��IRn�� As will be shown in Theorem ��� below MG��� ���
�
�n n
�ln ���
�is the sharp constant for all � � Now� let us compare the
function MG of ����� with the analogous one given by Davies� conditions� In hisframework� inequality ����� follows if��e�H�tg
��L��dx�
� kgkL��dx�eM�t�� t � �����
whereM�t� is a monotonically decreasing continuous function of t� In the presentcase� H� � �&�
e�H�tg � Kt � g� ������
where Kt�x� is the Gaussian ���t��n��exp
n� jxj�
�t
o� Then Young�s inequality gives��e�H�tg
��L��dx�
� kgkL��dx�kKtkL��dx� ������
and we obtain M�t� � �n�ln ��t� Thus
M����MG��� �n
���� ln �� � ������
and the function M��� is not optimal�
In the above example we were able to rewrite ����� as ����� with arbitrarily smallprincipal coe�cient �� For general potentials A�x� that satisfy �A��� the possibi�lity of doing so is deeply connected to the di�erence between hypercontractivityand ultracontractivity �see x of �Gro���� x� of �Dav����� As the heat kernelexample shows� the entropy approach of x� could lead to better constants�
��
��� Perturbation lemmata for the potential A�x�
In Section � we derived the convex Sobolev inequality ����� corresponding toFokker�Planck operators L� � �div�D�r� �rA�� that satisfy the Bakry�Emerycondition �A��� Next we will present two perturbation results to extend thisinequality to a larger class of operators�
First we shall consider bounded perturbations of the �potential� A�x�� Our resultgeneralizes the perturbation lemma of Holley and Stroock �HolSto���� �Gro��from the logarithmic entropy to all admissible relative entropies e� from De�ni�tion ����
Theorem ���� Let ���x� � e�A�x�� f���x� � e� eA�x� � L���IRn� with RIRn��dx �RIRnf��dx � � and
eA�x� � A�x� v�x��
a � e�v�x� � b �� x � IRn� ������
Let the symmetric locally uniformly positive de�nite matrix D�x� be such that theconvex Sobolev inequality ����� �with the admissible entropy generator �� holdsfor all f � L��d����Then a convex Sobolev inequality also holds for the perturbed measure f�� �Z
IRn
�
�f �
kfk�L��df���
� f���dx� ���� �
� �
�max�
b
a��b�
a�
ZIRn
f �
kfk�L��df���
����
f �
kfk�L��df���
�r�fDrf f���dx�
for all f � L��df��� � L��d����Proof� We introduce the notations
���x� ��f ��x�
kfk�L��d���
� ���x� ��f ��x�
kfk�L��df���
� � �������kfk�L��d���
kfk�L��df���
�
and because of ������ we have
�
b� � � �
a� ������
Below we shall need the estimate
������� ��
�������� �� ��� �����a���������� � �� � ��� �����b�
�
The �rst line follows from ���� � �see ������� and the second line follows from�������
We shall now �rst prove the assertion ���� � for the case � � �� In the followingchain of estimates we use� in this sequence� ������� ������� ������ ������� �����b���������Z
IRn
�����f���dx� �ZIRn
�������
� ����� ���� � ��� f���dx�
� ��ZIRn
�����f���dx� � b��ZIRn
��������dx�
� b���
�
ZIRn
f �
kfk�L��d���
�������r�fDrf���dx� ������
� b
a
�
�
ZIRn
f �
kfk�L��df���
�������r�fDrff���dx�� b
a
�
��
ZIRn
f �
kfk�L��df���
�������r�fDrff���dx�� b
a��
�
ZIRn
f �
kfk�L��df���
�������r�fDrff���dx��which is the result if �� � ���In the case � � we proceed similarly and �rst apply ������ to obtain�ZIRn
�����f���dx� � ZIRn
������� ����� ���� � ���f���dx� � �ZIRn
�����f���dx��������
Now we again use ������� ������ ������� �����a�� ������ and �nally obtain the resultZIRn
�����f���dx� � b�a
�
�
ZIRn
f �
kfk�L��df���
�������r�fDrff���dx�� �����
We remark that the �perturbation constant� in ���� � is not as good as the onein the proof of �HolSto��� �max� b
a�� b
�
a� versus b
a�� This is due to the fact that the
proof of Holley and Stroock exploits homogeneity properties of � and ��� in thecase ���� � � ln��� �� Thus� their original proof �with the constant b
a� can be
extended to entropy generators of the form ���� � �p� �� p��� ��� � p � ��but not to the general case of Theorem ����
Example ���� Set D�x� � I and consider the �double well potential� A�x� �c�jxj�� c�jxj��c��� �� Then the perturbation theorem ��� yields convex Sobolevinequalities as A�x� can be written as a bounded perturbation of a uniformlyconvex potential that satis�es �A���
��
Next we shall specialize our discussion to the one�dimensional situation and deriveconvex Sobolev inequalities under very mild assumptions on the potential A�x��In particular we shall prove that an appropriate choice of the di�usion coe!cientD compensates lack of convexity of the potential A�
Theorem ���� Let A � W ���loc �IR� be bounded below and satisfy ���x� � e�A�x� �
L���IR� withRIR���dx� � �� and let � generate an admissible relative entropy�
Then there exists a � and a function D � D�x� with D�x� � D�� x � IR�for some constant D� � such that�Z
IR
�
��
��
����dx� � �
��
ZIR
�����
��
�D
� ����x
� ���dx�� ������
� � L���IR� withZIR
�dx � ��
The idea of the proof is to construct a bounded function D�x� such that �A�D�satis�es �A��� To this end we need the following
Lemma ���� Let A�x� satisfy the conditions of Theorem ���� Then there existsa such that the ODE
�
�
D�x
D� ��Dxx
�
�DxAx DAxx � ������
admits at least one global solution that satis�es D�x� � D� � x � IR�
Note that the left hand side of ������ is the �� d version of the left hand side of�A�� such that Theorem ��� follows�
Proof� We transform D�x� � y�x�� and solve
yxx � Axxy Axyx � y� x � IR� ������
y�� � eA���� y��� � Ax��eA���� ���� �
The �possibly only local� solution of ������ satis�es
y�x� � eA�x� � eA�x�Z x
�
�e�A�z�
Z z
�
y�����d��dz � eA�x�� ������
Due to this upper bound� y�x� can only break down at a �nite x� if y�x�� �� Locally� y�x� can be obtained through a �xed point iteration starting withy��x� �� e
A�x� �
yk���x� � eA�x� � eA�x�Z x
�
�e�A�z�
Z z
�
yk�����d�
�dz ������
� eA�x���� ke�AkL��IR�
Z x
�
yk�����d�
� � k � �
��
Starting with y��x� � eA�x�� an iterative application of the estimate ������ yields
y�x� � �eA�x� ������
with
� � ��
��
�� � � �
��
�
r�
�� �
whenever � ���
By induction one shows that �yk� is decreasing� Hence� ������ converges toy�x� x � IR� From ������ we obtain D�x� � � �e�A� �� D�� x � IR� withA� denoting the lower bound of A�x��
Note that D � e�A satis�es ������ with � � Hence the couple �A�D � e�A�violates the Bakry�Emery condition �A��� nevertheless the convex Sobolev ine�quality ������ holds with D � e�A and � �
���due to the estimate ��������
Obviously� for A uniformly convex D can be chosen as a constant �cf� �A����We shall now show that for nonuniformly convex A� in general �A�� cannot besatis�ed with a uniformly bounded function D�
Lemma ���� Let A�x� satisfy the conditions of Theorem �����
a� If D�x� � D� �� x � IR� then limx��Ax�x� � ���b� If D� � D�x� � D� �� x � IR� then A�x� grows at least quadrati�
cally�
Proof� a� Using D � y� we rewrite the di�erential inequality �A�� as
yxx
y f � �Axy�x
for some f�x� � � Solving for A gives
Ax�x� ��
y�x�
�C
Z x
�
dz
y�z�
Z x
�
f�z�dz yx�x�
�� ������
for some C � IR� Since y is bounded on IR we have limx��yx�x� ��� Theresult follows from Z x
�
dz
y�z�� xp
D�
� x � �
��
b� Integrating ������ gives
A�x� � A��
�
�Z x
�
dz
y�z�
��
C
Z x
�
dz
y�z� ln
y�x�
y��
Z x
�
�
y�z�
Z z
�
f���d�dz�
and the result follows from the boundedness assumptions on D�
We now �nish our discussion of perturbation results on convex Sobolev inequali�ties with an example� It demonstrates that the di�usion coe!cient D constructedin the proof of Lemma ��� in many cases has a much stronger growth at x � ��than necessary to satisfy the Bakry�Emery condition �A���
Example ��� Let A � C��IR� satisfy A�x� � cjxj�� for jxj L and � � ��Then a pair �D� � satisfying �A�� can be constructed such that
D�x� �
� �c�� c
�� x
����� � x L��c�� c
�� jxj���
��� x �L�
� �����
for some c� � IR c� L� L� The construction of D proceeds as follows�On the interval ��L�� L��� where L� will be determined later we choose D � y��where y is the solution ���� � of the IVP ������� Outside of this interval weextend D by ������ in a C��way thus �xing c��� �in dependence of and L��� Bya perturbation argument around � one easily sees that can be chosensmall enough and L� large enough such that �yx�x���L�� � and such that �A��holds for jxj L��
��� Poincare�type inequalities
As an application of Theorem ��� we shall now derive Poincar�e�type inequalities�
Remark ���� Poincar�e�type inequalities on bounded uniformly convex domainsare readily obtained from convex Sobolev inequalities� For simplicity�s sake letB � fx � IRnj jxj �g be the unit ball in IRn and let �� � ���x� � � ���x� � �on B be the density of a probability measure on B� We de�ne the C��function'A� � 'A��jxj� on IRn for � by
d�
dr�'A��r� �
��� r ���� r �
� 'A��� �d
dr'A��� � �
We set
A��x� �
�'A��jxj�� jxj�
�� ln ���x�� x � B
'A��jxj�� x �� B
��
and
����x� � exp��A��x���ZIRn
exp��A��y��dy�
Obviously
����x������
����x�� x � B� x �� B �
Since A� �� � ln ��� is an L�� perturbation �uniformly as �� � of the uniformlyconvex function 'A��jxj� which satis�es
�� 'A�
�x�� I on IRn�
we can apply the perturbation result Theorem ��� and obtain the convex Sobolevinequality �with D � I�Z
IRn
�
�vR
IRnvd���
�d��� � c
ZIRn
����
vRIRnvd���
� jrvj��RIRnvd�����
d���
for all admissible entropies � and all functions v � L���IRn� d���� �v � L��IRn� d����if � is quadratic on IR�� Here c is independent of ��
Passing to the limit �� gives the Poincar�e�type inequalityZB
�
�vR
Bvd��
�d�� � c
ZB
����
vRBvd��
� jrvj��RBvd����
d��
for all v � L���B� d��� �v � L��B� d��� if � is quadratic on IR�� In the latter case���� � �� � ��� with �� � �
vol�B�we obtain the classical Poincar�e inequalityZ
B
�v � �
vol�B�
ZB
vdx
��
dx � �cZB
jrvj�dx� v � L��B��
Obviously the above limit argument can easily be carried over to general uniformlyconvex domains�
��� Sharpness results
We now turn to the analysis of the saturation of the convex Sobolev inequalities�i�e� we shall answer the question for which function � the inequality ����� becomesan equality� In particular we shall �nd necessary and su!cient conditions onthe entropy generator � and on ��� which imply the existence of an admissible
�
function � � �� such that ����� �under the assumption D � I� becomes anequality� We remark that this question was completely answered in �Car��� forthe Gross logarithmic Sobolev inequality �i�e� ���� � � ln� � � �� �� � Ma�by a technique di�erent from the one presented in the sequel� The same problemwas treated in �Tos��a� and �Led��� using a method which we shall generalizebelow�
At �rst we observe that the derivation of the convex Sobolev inequalities givenin Section � is based on writing the entropy equation
d
dte����t�j��� � I��e�t�j���� ������
the equation for the entropy production
d
dtI����t�j��� � ���I����t�je�� r����t�� ������
and proving r����t�� � � Explicitly� we obtain by inserting ������ into the righthand side of ������ and by integrating with respect to t�
e���I j��� � � �
��I���I j���� �
��
Z �
�
r����s��ds ������
where ��t� in the Fokker�Planck trajectory which �connects� the initial state �Iwith the steady state ��� r� � then gives the convex Sobolev inequality
e���I j��� � �
��jI���I j���j�
which becomes an equality i�Z �
�
r����s��ds � �� r����t�� � a�e� in IR�t � ������
The precise form of the remainder r� can easily be extracted from the proof ofLemma ���� Assuming henceforth
D � I ���� �
we obtain
r���� �
ZIRn
��IV�eA��juj� ������eA��u��u
�xu �����eA��
nXl�m�
��ul�xm
��
�e�Adx
�
ZIRn
����eA��u����A
�x�� �I
�u e�Adx� ������
��
where we recall u � r�eA��� Obviously� r���� � if u � � which �takinginto account the normalization� implies � � �� and gives the trivial case ofequality in the convex Sobolev inequality� Thus� assume u � � Then� using theadmissibility conditions ������ for the entropy generator � and �A��� we concludethat r���� � holds i� the subsequent four conditions are satis�ed�
�����eA��� ��
�����eA���IV�eA��� �����a�
juj��
nXl�m�
��ul�xm
��
� ��
� ju��u�xuj� �����b�
����eA��u��u
�xu � ������eA��juj�� �����c�
u����A
�x�� �I
�u � � �����d�
Since ��t� �� eA��t� satis�es ������ we have by the maximum�minimum principle
infIRneA�I � eA�x���x� t� � sup
IRneA�I
a�e� in IRn � IR�t � We conclude from �����a��
Lemma ���� If the Sobolev inequality ������ with D � I becomes an equality for� � �� then ���� � ���� or ���� � ���� holds for � � �infIRn eA�I � supIRn eA�I�where � and � are given by ���� a� and ���� c� resp�
Nontrivial saturation can therefore only occur for the #minimal$ and #maximal$entropies�
At �rst we investigate the case of the maximal �quadratic� entropy� Note thatnow any � � L��IRn� dx� is admissible� Positivity is not required�Theorem ���� The convex Sobolev inequality ������ with D � I and ���� ��� � ��� becomes an equality i� the following two conditions hold�
�i� there exist Cartesian coordinates y � �y�� � � � � yn�� � y�x� on IRn such that
for some � � IR
A�x�y�� ���y�� �y� B�y�� � � � � yn�� ������
��
�ii� � satis�es for some � � IR�
��x�y�� � �� �y��e�A�x�y��� ������
Proof� Since ���� � we conclude from �����b� and �����c���ul�xm
� % l� m � �� � � � � n �����
�u � +�� Thus u�x� t� � rx�eA�x���x� t�� � C�t�� where C�t� is constant in
x� Then ��x� t� � �C�t� � x C��t�� e�A�x� follows with C� real valued� Inserting
into the Fokker�Planck equation gives ,C � x ,C� � �rA � C and ,C � ���A�x�C
follows� �����d� gives C�t����A�x�
� �I�C�t� � and since ��A
�x�� �I �
we conclude ��A�x�C�t� � �C�t�� We obtain ,C � ��C and C�t� � C�e
���t�
Therefore��A�x�
� �I�C� � and we have rA�C���C� �x � � � IR� We insert
C�t� � C�e���t into ,C �x ,C� � �rA �C and �nd C��t� �
���e���t C�� C� � IR�
This gives
��x� t� �
�C� � x �
�
�e��t�A�x� C�e
�A�x�
and t�� implies C� � �� Summing up� we found� for some C� � IRn� � � IR�
��x� t� �
�C� � x �
�
�e���t�A�x� e�A�x� �����a�
and
r�A� ��jxj�� � C� � �� �����b�
Note that C� � implies � � and ��x� t� � e�A�x� follows� We therefore assumeC� � from now on�Set �� �
C�
jC�j � choose orthonormed vectors ��� � � � � �n � f��g and de�ne thechange of coordinates x � y by x � y��� � � � yn�n� We have rxf�x� � C� ��f�x�y���y�
jC�j and thus
A�x�y�� ���y��
�
jC�jy� B�y�� � � � � yn�
follows from �����b�� and this �nishes the proof�
Next we treat the #minimal$ entropy case�
��
Theorem ����� The convex Sobolev inequality ������ with D � I and ���� �� ln� � � � becomes an equality i� the following two conditions hold�
�i� there exist Cartesian coordinates y � �y�� � � � � yn� � y�x� on IRn such that
for some � � IR
A�x�y�� ���y�� �y� B�y�� � � � � yn��
�ii� � satis�es for some � � IR
� � exp
��A�x�y�� �y� � �
�
�� ��
�
�� ������
Proof� We set z � r ln��eA� in ������ and calculate �using the speci�c form of���
r���� � �
ZIRn
�
nXl�m�
��zl�xm
��
dx �
ZIRn
�z����A
�x�� I
�zdx�
Assume z � � Then �z�x� follows and z�x� t� � C�t� � IRn� This gives��x� t� � C��t� exp�C�t� � x� A�x��
with C��t� � Inserting into the Fokker�Planck equation and proceeding as inthe proof of the previous Theorem implies
��x� t� � exp�C� � xe��t �
�e��t � A�x�
�������
where � and C� satisfy �����b�� Setting t � in ������ proves the assertion�
Note that � in ������ is obtained by a shift in the y��coordinate of �� �cf� �Car�����
In one dimension �n � �� or for constant matrices D the analogous result tothe Theorems ��� and ��� is obtained by the coordinate transformation of Re�mark ���� For D�x� � I� however� nontrivial saturation of the convex Sobolevinequality ����� is not always possible for any A�x�� even for scalar di�usionD�x� � D�x�I� the analogue of ����� implies an integrability condition on D�x��which is not satis�ed in general�
� A generalized Csisz�ar�Kullback Inequality
We shall now be concerned with proving generalized Csisz�ar�Kullback inequali�ties� which are estimates for the L��distance of two functions in terms of their rela�tive entropy� These inequalities have at least a � years history in probability and
��
information theory �Csi��� Csi��� Csi��� Kul �� KuLei �� Per �� Per� � BaNi����In �Csi��� �Theorem ���� page ��� and Section �� page ���� the following resultwas obtained�
Theorem ���� Let e� � ����� IR be bounded below convex on ���� strictlyconvex at � with e���� � � Then there exists a function W
e� � IR � ���� suchthat
�� We� is increasing�
�� We��� � �
�� We� is continuous at �
�� For all non�negative u � L��)�S� �� �where �)�S� �� is a probability space�with
R�u d� � � the Csisz�ar�Kullback inequality holds�
ku� �kL��d�� � We��e e��u��� �����
where ee��u� ��
R�e��u� d��
For the generating function e���� � � ln� � � �� e�g�� it is known �Csi��� thatWe��c� �
p�c� �����
By a rescaling argument the inequality ����� can be extended to functions u �L��d��� u � � which are not necessarily probability densities� Excluding the caseu � � we replace u by u��R
�u d��� apply ����� and multiply by
R�u d� to get
����u� Z�
u d�
����L��d��
��Z
�
u d�
�W�
ee��u�
��
where ���� � e��R�ud� ���
Throughout this Section we shall make use of the following assumptions andnotations�
B�� �)�S� �� is a measure space and � is a probability measure on S�i�e� ��)� � �� � � �
B�� � � J � IR is a strictly convex� continuous� non�negative function onthe �possibly unbounded� interval J � which has a non�empty interior J��If � �� inf J �� J � then lim�� ���� � �� If � �� sup J �� J � thenlim�� ���� ���
�
For functions u � L��d�� �� L��)% d�� we de�ne the entropy
e��u� ��
�����Z�
��u� d�� ���u�� if u�x� � J ��a�e��
� else�
�����
with the notation �u� ��R�u d��
Remark ���� Since � is continuous on J the mapping x �� �� � u��x� is S�measurable whenever u is S�measurable� Furthermore we have due to Jensen�sinequality for all u � L��d�� with u�x� � J �� a�e�Z
�
��u� d� � ��Z
�
u d�
��
Thus the right�hand side of ����� has a well�de�ned value in �� ���Remark ���� a� In contrast to De�nition ����� the generating function � doesnot have to satisfy the normalization ���� � ����� � �b� The required continuity of � as well as the assumed behaviour of ���� as �tends to the boundary of J deserve a few more comments� Let -� � J � IR beconvex on the interval J which has a non�empty interior� Then -� is continuouson J� and possible points of discontinuity are contained in �J J� If � � IRand if ���� remains bounded as � approaches � then we can �re��de�ne -���� ��lim���
-����� We get a convex function less or equal to the original one �i�e� theentropy decreases� which is continuous at �� The same procedure applies incases where � is �nite and ���� remains bounded as � approaches �� The cases� � �� or � �� are discussed in c��c� Not every strictly convex continuous function -� � J � IR on the �possiblyunbounded� interval J with non�empty interior satis�es B��� However we cannormalize the generating function -� as in Remark ���� Due to the strict convexityof -� there exist for each �� � J� a constant b � IR such that the function
���� �� -����� -������ b�� � ���
is non�negative on J� Furthermore we have for all u � L��d�� the equalitye��u� � e ���u� which shows that � and -� generate the same entropy�
Remark ���� Let us assume that ) � IRn n � IN is a Borel set S is the Borelsigma algebra on ) and � is a probability measure on S which is absolutelycontinuous with respect to the Lebesgue measure� In this case the Radon�Nikodymderivative of � with respect to the Lebesgue measure exists�
g�x� ��d��x�
dx�
��
To keep things simple let us assume that g�x� for almost all x � )� Nowconsider a function f � L��)� dx� which satis�es
u�x� ��f�x�
g�x�� J a�e� on )�
Then e��u� is � as in the previous sections � the relative entropy of f w�r�t� g�
e��u� � e��f jg� �Z�
��f�g�g dx� ��Z
�
f dx
��
In contrast to De�nition ��� howeverR�f dx needs not to be normalized here�
In the case of ���� � �ln��� � and f�x� � with R�f dx � � the inequalities
����� and ����� combine to �������
The subsequent analysis aims at generalizing the above entropy estimates� Inparticular we will establish a generalized Csisz�ar�Kullback inequality of the form����u� Z
�
u d�
����L��d��
� U
�e��u��
Z�
u d�
�� �����
where the function U only depends on � and u belongs to L��d��� We shallpresent a transparent �and almost elementary� construction of U and thus extendthe results of the above cited references to a larger class of functions u �u does nothave to be normalized or non�negative here�� Then we shall prove the optimalityof the estimate and construct U explicitly for certain convex functions �� Theexplicit construction of U is somewhat lengthy and thus deferred to the Appendix�
In the sequel we shall need the following ��dependent constants� For any � � J�we de�ne�
Q����� �� limz��
��� z�� ����z
� ����� if � ���
Q����� �� limz��
��� � z�� ����z
� ����� if � � ���
c���� ��
�������������
����� ���� �� � ��Q����� if �� ���� � IR� � �������� ���� �� � ��Q����� if �� ���� � IR� � � ���
�� � ������ �� � ������� � � � ���� if �� �� ����� ���� � IR�
� else�
��
�with the convention � � �� for all � �� and
Usup��� ��
�������������� � �� if � � IR� � ����� � �� if � � ��� � � IR
��� � ���� � ��� � � if �� � � IR� if � � ��� � ��
We now collect the essential properties of U in the following theorem% its elemen�tary proof is given in the Appendix�
Theorem ���� Under the assumption B�� there exists a function U � ���� �J � IR such that
P�� U is continuous�
P�� For all � � J� U�� �� � �P�� For all � � J� The mapping c �� U�c� �� c � ���� is increasing�P�� For all � � �J J and all c � ���� the identity U�c� �� � holds�
P�� For all � � J�� The mapping c �� U�c� �� c � ���� is strictly increasingon �� c������
P � For all � � J�� limc��
U�c� �� � Usup����
P�� For all � � J� and all c � �c�������� U�c� �� � Usup����
We note that �J J �see P��� may be empty� In P�� we use the convention����� � �� The main result of this Section is�Theorem ��� Assume B�� � B��� Then we have for all u � L��d�� withe��u� �� ����u� Z
�
u d�
����L��d��
� U
�e��u��
Z�
u d�
�� ��� �
Proof� We �rst shall introduce some abbreviations� Recall that �u� ��R�u d��
Since e��u� � we have u�x� � J ��a�e� Hence �u� � J � We set)�� �� fx � ) � u�x� � �u�g� )� �� fx � ) � u�x� �u�g�
and de�ne
w �� u� �u� � a ��Z���
w d� � � �� ��)�� ��
��
We note that � � �� �� and ��)�� � � � �� SinceR�w d� � � we haveR
��w d� � �a and����u� Z
�
u d�
����L��d��
�
Z�
jwj d� �Z���
w d��Z��w d� � �a�
�This factor � also appears in the de�nition of U�� We proceed by a case�distinction� The �rst case is probably the most interesting one�
Case �� � � �� If �u� � �J J � then we will get u � �u� and therefore � � �� Thisis a contradiction since we have assumed � �� Hence �u� � J�� Furthermore weget from the de�nition of a� �u� and ��
a � minf��� � �u��� ��� ����u�� ��g�
and therefore
a � Usup��u��
���� � �u����u�� ��
�� � �
�a
� � �u� � ��a
�u�� �� �� ���
Now we apply Jensen�s inequality�
� e��u�
�
Z�
���u�� � ��u��� d�
�
Z�
�� �w �u��� � ��u��� d�
�
Z���
�� �w �u��� � ��u��� d� Z���� �w �u��� � ��u��� d�
� ���u�
a
�
�� ���u��
� ��� ��
��
��u�� a
�� ��� ���u��
�
� a
����u� a
�
�� ���u��a�
���u�� a
����� ���u��
a���
��� -���u�� a� ��
� inff-���u�� a� �� � � � J��u�� a� �� ��g�
��
where
J��u�� a� ��
�����������������������������
�a
� � �u� � ��a
�u�� ��
if �� � � J�a
� � �u� � ��a
�u�� ��if � �� J� � � J�
a
� � �u� � ��a
�u�� ��
if � � J� � �� J�a
� � �u� � ��a
�u�� ��
if �� � �� J
We have to distinguish between three cases�
Case �a� a Usup��u����� In this case the interval J��u�� a� �� �� has a non�empty interior� Due to the fact that ���� � ��� � �if � � J� and ���� � ������if � � J� we have
e��u� � infn-���u�� a� �� � � � J��u�� a� �� ��
o� inf
�-���u�� a� �� � � �
�a
� � �u� � ��a
�u�� ����
and therefore with the notations of the proof of Theorem ��
I�e��u�� �u�� � a �ku� �u�kL��d��
��
which gives due to the de�nition of U
U�e��u�� �u�� � ku� �u�kL��d���Case �b� a � � This case needs not to be considered here because a � impliesu � �u� and therefore � � �� But this is a contradiction because it is assumedthat � ��
Case �c� a � Usup��u����� Since a � IR we have � � IR or � � IR� If � � IR and� � � we will get the contradiction � � �� � � �� and if � � �� and � � IRwe will get the contradiction � � ��� �� � �� Hence� the only remaining case is�� � � IR� We get � � ��u�� ����� � �� which gives
e��u� � ��u�� ������ �� � �u������� � � � ���u��
� ��u�� ������� �� � �u����� �� � � � ���u��
� c���u���
and therefore by the de�nition of U and Usup��u��
U�e���u��� �u�� � U�c���u��� �u�� � Usup��u�� � �a � ku� �u�kL��d���
Case �� � � �� In this case we easily get u � �u�� Hence� e��u� � � ku ��u�kL��d�� and therefore
U�e��u�� �u�� � U�� �u�� � � ku� �u�kL��d���
We shall now discuss inequality ������
��� Optimality
A natural question in connection with ����� is the following� Is it possible toimprove ������ i�e� is there a function U� � f�c� �� � � � J�� c � �� c������J�g �forreasons explained in Remark ��� below we exclude the cases c � �c������� and� � �J J here� such that
�� for all u � L��d�� with e��u� �� ku� �u�kL��d�� � U��e��u�� �u���
�� there exists a u� � L��d�� with e��u�� � andU��e��u��� �u��� U�e��u
��� �u��� .
Under the assumption B�� �see below� the answer to this question is no�
Theorem ���� Assume B�� B�� and furthermore
B�� For all � � �� �� there exists a set S� � S such that ��S�� � ��
Then for all � � J� and c � �� c����� there exists a sequence �um�m�IN in L��d��such that
�� For all m � IN � �um� � ���� For all m � IN � e��um� � c��� lim
m��kum � �um�kL��d�� � U�c� ���
�
Proof� Let us consider the case c c���� �rst� In this case there exists a�uniquely determined� a � ��Usup��� such that
c � inf���a���������a�������
-���� a� ���
where -� is de�ned as in the proof of Theorem ���� Clearly� �a � U�c� ��� Wenote that the mapping � �� -���� a� �� is not constant and the mapping a ��inf���a���������a������� -���� a� �� is increasing� Furthermore� -� is continuous� Wecan therefore choose a sequence ��m�m�IN in �� �� and a sequence �am�m�IN in�� a� such that -���� am� �m� � c and limm�� am � a� Now we choose a sequence�Sm�m�IN in S such that ��Sm� � �m for all m � IN � We set for all m � IN
um � ) � IR
x ���
�am��m� � if x � Sm��am���� �m�� � if x � ) n �
We easily verify that �um� � � and e��um� � c� Furthermore�
limm��
kum � �um�kL��d�� � �a � U�c� ���
This �nishes the proof for the case c c����� If c � we choose um � ��which gives �um� � �� e��um� � � c and
kum � �um�kL��d�� � � U�� �� � U�c� ���
Remark ���� In Theorem ��� the cases � � �J J and � � J� with c ��c������� c���� � are not included� Indeed if � � �J J then we wouldhave u � �u�� Hence e��u� � and in this case the estimate ����� is optimaltoo�
ku� �u�kL��d�� � � U�� �� � U�e��u�� �u���
If � � J� and e��u� � c���� we will trivially get for all u � L��d�� with u�x� � J��a�e� and �u� � �
ku� �u�kL��d�� � Usup��u�� � U�c���u��� �u�� � U�c� �u���
We note that this inequality is strict in cases where � � �� or � ���
��� An extension of the inequality
A closer screening of the proof of Theorems �� and ��� shows that the core of allestimates is the fact that for all � � J� and all a � ��Usup��� the strict estimate
inf���a���������a�������
-���� a� ��
�
holds� For �xed �� � J� this is the case i� � is strictly convex at ��� i�e� i� thereexists a constant b � IR such that the strict estimate
���� ����� b�� � ���
holds for all � � ��� ��� � � ��� We can therefore extend Theorems �� and ���in the following way�
Theorem ���� Assume B�� and
B��� '� � J � IR is a convex continuous non�negative function on the �possiblyunbounded� interval J which has a non�empty interior J�� If � �� J thenlim��
'���� � �� If � �� J then lim��'���� � �� Furthermore assume
that '� is strictly convex at �� � J��
Then�
a� There exists a function U� � ����� IR such that
P�� U� is continuous�
P�� U��� � �
P�� U� is increasing�
P�� U� is strictly increasing on �� c�������
P�� limc��
U��c� � Usup�����
P � For all c � �c��������� U��c� � Usup�����
b� Let u � L��d��� If e��u� � and �u� � �� then����u� Z�
u d�
����L��d��
� U��e ��u���
We note that an optimality result as Theorem ��� also holds for U� �
As an example consider the function '���� � j�j which is strictly convex at �� � �In this case we get ej�j�u� �
R jujd� and U��c� � c which trivially gives for allu � L��d�� with �u� �
kukL��d�� � U��ej�j�u�� � U��
Zjujd�� �
Zjujd��
�
��� Comparison with the classical Csiszar�Kullback ine�
quality
Inequality ��� deepens the insight into the classical Csisz�ar�Kullback inequalityin four ways� Firstly� inequality ��� is valid for all functions u with e��u� ��i�e� there is no normalizing condition
Rud� � � assumed and u may be negative�
Secondly� we have an optimality result for the function U� Thirdly� it has beenproven that U is strictly increasing on �� c������ Fourthly� the de�nition of Uis rather straightforward and requires no measure�theoretic background but onlyJensen�s inequality�
��� An example
The calculation of U involves a minimizing procedure� If � is di�erentiable thiswill lead to the problem of �nding roots of transcendental equations� Hence thereis no hope for an explicit presentation of U in general�
In some cases� however� the function U can be found explicitly� at least for specialvalues of �u� �
R�u d�� If ���� � j� � �j�p� p � ������ on J � IR� we have for
all admissible u with �u� � ��
ku� �kL��d�� ��Z
�
�juj�p � �� d�����p
�
In the situation described in Remark ��� this becomes with the additional requi�rement
R�f dx � ��
kf � gkL��dx� ��Z
�
jf � gj�p g���p dx����p
�
For the special case of p � � and �u� � � the above two inequalities read�
ku� �kL��d�� �sZ
�
�u� � �� d��
and respectively�
kf � gkL��dx� �sZ
�
�f � g�� g�� dx�
�
��� Asymptotic behavior of U as e��u�� �
As mentioned before U can in general not be expressed in terms of elementaryfunctions� For �small� values of e��u�� however� one can obtain an asymptoticexpansion of U� Let us recall the following result of �Csi���� If � � C��J�� � � J�and ������ � then the Cszis�ar�Kullback inequality holds�� �K��� � u � L��d��� u � � �u� � � � If e��u� � K� then
ku� �kL��d�� � K�
pe��u��
�����
In �Csi��� the investigations are specialized to the case ���� � � ln�� A strongerresult holds in this case�
For all u � L��d�� with u � R�u d� � ��
ku� �kL��d�� �s�
Z�
u lnu d��
We note that ������ was essential to get ������ Otherwise one can� in general�not get an estimate of the form
ku� �kL��d�� � K �e��u��� � K� � � �����even for small values of e��u� �to keep things simple we consider only the caseR�u d� � ���
Example ����� Let ) � �� �� d� � dx� Let
� � IR� IR � � ��Z
�
��s� ds
with
� � IR � IR
� ��
�������������� � �� � � � ���� ���e������ � � � �� ��� � � � ��e������ � � � ��� ���e���� � �� � � � ��� ���
Certainly � is bounded below strictly convex and belongs to C��IR� with ���� ������ � ������ � � For k � IN let
uk � �� �� � IR
� ���� ���k� � � � �� ������ ���k� � � � ����� �� �
For all k � IN we have uk � L��dx� uk � R�uk dx � � and kuk � �kL��d�� �
��k� Furthermore for all k � IN �
e��uk� ��
�
��
�k �
k
� �
�k � �k
���
Hence
limk��
kuk � �kL��d�� � limk��
e��uk� � �
Now let � � ����� Then a simple application of the de l�Hospital rule gives
limk��
kuk � �k���L��d��
e��uk��
�
��� e�lim��
a�����e�� ���
The core of this example is the fact that all derivatives of � vanish at ��
However� if � is m�times �m � �� continuously di�erentiable in a neighborhoodof �u� � J� with
����u�� � � � � � ��m�����u�� � � ��m���u��
�due to the convexity of �� ��m���u�� has to be non�negative�� we can use theobservation that for all �u� � J�� all su!ciently small a � ���� and � � �a�����u��� �� �a���u�� ���� �see the proofs of Theorems �� and �����
���u� a
�
�� ���u��a�
���u�� a
����� ���u��
a���
� � ��u� a�� ���u��a
� ��u�� a�� ���u��
a�
�����
Now we can use a Taylor�expansion argument to obtain
if m is even���u� R
�u d�
��L��d��
�� o���� � �
�m+ e��u�
���m��R
�u d�
����m
�
if m is odd���u� R
�u d�
��L��d��
� o��e��u����m�����
�����
as e��u�� �
The upper estimate can be re�ned in the following cases�
�
Case �� ���u�� � ����u�� � � �����u�� and � has a logarithmic subentropy�Then � as it is shown in Section � � the term o��� can be ignored and the estimate
ku� �u�kL��d�� �s� e��u�
�����u�������
holds�
Case �� ���u�� � ����u�� � � �����u�� � and ���� � in a neighborhood of�u� �
R�u d�� Then the term o��� is nonnegative and we obtain ����� for all
admissible u with �u� � J� as e��u�� � We note that this estimate holds for alle��u� � ���� whenever � � C��J� with �����u�� and ���� � on J � As anexample consider ���� � e � e �� We have
ku� �kL��d�� �s�
Z�
�eu�� � �� d��
for all admissible u with �u� � ��
Let us keep the assumption that � is three times di�erentiable in a neighborhoodof �u� with �����u�� � Then the estimate ����� is in general not optimal�Consider e�g� ���� � � �ln� � ln�u� � ��� where �u� � ����� We obtain from����� for all admissible u�
ku� �u�kL��d�� �� o���� �s��u�
�Z�
u lnu d�
�� ��u�� ln�u��
where due to ���� � the term o��� is negative� while on the other hand �asmentioned above� we have for all admissible u with �u� � ��
ku� �kL��d�� �s�
Z�
u lnu d��
The reason for this is the fact that the Taylor series argument used to obtain����� provides in general only an upper estimate for U� Whenever this estimate isnot sharp � e�g� for ���� � � ln��� � estimate ����� can be improved� In general�however� this improvement requires a more detailed discussion of �� Let us recallthat in Section � such a detailed discussion was actually carried out� If � admitsa logarithmic�type superentropy� then estimate ����� will hold for all values ofe��u� ��
If the function � is not di�erentiable at �u� �R�u d� � J�� then �����u�� �
�����u�� and we get from ���������u� Z�
u d�
����L��d��
� � e��u�
�����u��� �����u��
�
as e��u�� �
We can draw the following conclusion from this discussion� The more derivativesof ���u�� vanish �where �u� � J��� the slower u converges to �u� in L��d�� ase��u�� �
�� Passing from weak to strong convergence in L��d��
Inequality ����� allows to pass from weak L��d���convergence to strong L��d���convergence in the following sense� Given a sequence �uk�k�IN in L��d�� and aK � IR with
i�R�uk d�� K as k �� �this is� e�g� the case if uk u as k �� weakly
in L��d��� where u � L��d����ii� There exists a strictly convex and bounded below function � � J � IR �Jis not necessarily open� such that�
ii�a� For all su!ciently large k � IN � uk�x� � J ��a�e��ii�b� lim
k��
Z�
��uk� d� � ��K��
Then
uk � K strongly in L��d�� as k���
The non�trivial fact is that no growth conditions on � are imposed� Considere�g� the case where ���� � e� and let �hk�k�IN be a sequence in L��dx� and leth � L��dx� such that h�x� for almost all x � )� Then by setting d� � h�x� dxwe see that
limk��
Z�
hk dx �
Z�
h dx
together with
limk��
Z�
e�hkh��h h dx � e�h� �h�
implies hk � h strongly in L��dx� as k���
�
��� The optimal inequality for the logarithmic entropy
Now we consider the case ���� � �ln��� � with J � ���� � ��� ��� ��� � �and �u� � �� We recall that in this case we have for all u � L��d�� with e��u� �the estimates
ku� �kL��d�� � U
�Z�
u lnu d�� �
�� �����
�from ������ and
ku� �kL��d�� �s�
Z�
u lnu d�� ������
�Csisz�ar�Kullback inequality�� In the following plot �Figure �� we compare thefunctions U�c� �� �solid line� and
p� c �dashed line��
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
c
Figure �� U�c� �� �solid� andp� c �dashed� for ���� � �ln� � � �
We observe that for all u � L��d�� with �u� � � and u�x� � �� a�e� the estimate
ku� �kL��d�� � � Usup��� � ���� �
holds� Hence the estimate given by ������ provides no information when e��u� ��� From ������ however� one gets a non�trivial bound of ku��kL��d�� for all �nitevalues of e��u��
�
��� Entropy�type estimates on kf � gkL��d��
In this subsection we investigate possibilities to estimate kf � gkL��d�� in termsof
e��f jg� �Z�
��f�g�g d��
where we make the assumptions
B�� �)�S� �� is a measure space and � is a probability measure on S�i�e� ��)� � �� � � �
B��$ � � J � IR is a convex� continuous� non�negative function on the �possiblyunbounded� interval J � � � J�� � is strictly convex at �� and ���� �� If � � inf J � J � then lim�� ���� � �� If � � sup J � J � thenlim�� ���� ���
B�� g � L��d�� is positive for ��almost all x � ) and R�g d� � ��
B�� f � L��d�� satis�es f�x��g�x� � J for ��almost all x � )�
If we additionally assume that � is strictly convex� then we obtain from Theorem��
kf � gkL��d�� � U �e��f jg�� ���f ��� �f �� j�f �� �j � ������
with the notation �f � �R�f d��
������ can be applied whenever e��f jg� and �f � are known� The question ariseswhether kf � gkL��d�� can be estimated just in terms of e��f jg�� We will give thesurprisingly simple� a!rmative answer to this question below� To prepare thediscussion we shall introduce some abbreviations �rst� We de�ne
J� �� f� � ���� � � � � Jg � J� �� f� � ���� � �� � � Jg�
and set
�� � J� � ����� �� ����� �� ��� ��
��� � J� � ����
� �� ����� �� ���� �� �
Clearly � are continuous� non�negative� convex functions which are strictly con�vex at � Their respective epigraphs
epi��� � f��� z� � z � ����g
�
are due to B���� closed and convex subsets of IR�� Following �EkTe��� it is easyto see that the closure of the convex hull of epi���� � epi���� is given by
co �epi���� � epi����� � epi�co���� �����
where the function co���� ��� � J� � J� � ���� is the envelope of all a!nefunctions lower or equal to �� respectively� Some properties of co���� ��� arecollected in the following lemma�
Lemma ����� Assume B���� Then�
a� co���� ��� is convex�
b� co���� ��� is continuous�
c� co���� ��� is strictly increasing�
d� co���� ����� � and co���� ������ for all � � J� � J� � �
The proof of Lemma ���� is deferred to the appendix�
Remark ����� Due to a� and d� of Lemma ���� the function co���� ��� isstrictly convex at � One may ask whether co���� ��� is strictly convex onJ� � J� whenever �� and �� are strictly convex� The following example gi�ves a negative answer to this question� Let �� � ���� � IR� � �� �� and let�� � ����� IR� � �� ��� Then it is easy to see that
co���� ��� � ���� � IR
� ��
����������� � � � �� �
�
p��
�p�
�� � ��
��� � � ��
�
p�� �
�
p��
�� � � � ���
p����
�
i�e� co���� ��� is not strictly convex�
We set
e�� �� sup�J� J�
co���� �������
The main result of this subsection is
Theorem ����� Assume B�� B��� and B��� Then�
��
a� For all f � L��d�� satisfying B�� the following estimate holds�
kf � gkL��d�� ���co���� ������ �e��f jg�� � e��f jg� e��
maxf�� �� � � �g � e��f jg� � e��
b� If there exists for all � � �� �� a set S� � S such thatRS�g d� � � then
there exists for all � � J� � J� a sequence �fn�n�IN in L��d�� such thatB�� is satis�ed for all n � IN and
kfn � gkL��d�� � � for all n � IN �
� � limn��
�co���� ���
����e��fnjg���
The proof of Theorem ���� is deferred to the appendix� Part b� of Theorem���� is an optimality result �the knowledge of e��f jg� does not allow to concludemore about kf � gkL��d�� than stated in Theorem ����� which applies� e�g� forthe n�dimensional Lebesgue�measure d� � dx� The calculation of co���� ��� isvery simple in cases where �� � �� with J� � J� �or �� � �� with J� � J��holds� In such situations we have
co���� ��� � �� �or co���� ��� � ����
As an example consider ���� � � �ln� � �� �� Then J� � �� ��� J� � ����with
����� � �� �� �ln�� ��� �� � � ��� �� �ln��� ��� �� � � ������for all � � �� ��� Thus co���� ������ � �� �� �ln�� ��� �� � and we obtainProposition ����� Assume B�� B��� and B��� Then for all f � L��d�� withf � the estimate
�� kf � gkL��d����ln�� kf � gkL��d���� �� ��Z�
f ln�f�g� d�
Z�
�f � g� d�
holds�
� Nonlinear Model Problems
We conclude this paper with an application of the theory presented in the previousSections to nonlinear Fokker�Planck type equations�
��
��� Desai�Zwanzig type models
Firstly� consider the following model describing the interaction of a system ofcoupled oscillators in the thermodynamic limit�
�t � Ddivx�rx� rxA�x� �% ��t����� t � � ��a�
��t � � � �I�x� ��� � ��b�
where x � IRn and the parameter vector � � ���� � � � � �M� � IRM � D is a positivedi�usion constant� � presents noise in the oscillator interaction and the oscillatorensemble potential A is given by
A�x� �% ��t�� � ��c�
��
D
�W �x�
�
�jz��t�� xj� � x �
MXl�m�
s��m�t�Elm�l �
�
MXl�m�
Elms��l�t� � s��m�t���
Here W is the single oscillator potential �which we assume to be purely determi�nistic� i� e� without noise�� � is a nonnegative parameter �interaction strength�and E � �Elm�l�m����� �M is a symmetric real matrix� on whose largest eigenvaluee� we will impose conditions later� We have
�C�� E � e�I
� in the sense of p�d� matrices�� The nonlinearity in the model stems from theoccurrence of the moments
z��t� ��
ZIRM�
ZIRnx
x��x� �� t�dxdP ���� � ��d�
s��m�t� ��
ZIRM�
ZIRnx
�mx��x� �� t�dxdP ���� � ��e�
The probability measure P represents the distribution of the noise ��
For the following we shall assume
�I � L���IRM� � IRnx% dxdP ���� andZIRM�
ZIRnx
�IdxdP � ��
For more information on the physics of the problem� an extensive list of referencesand a preliminary asymptotic analysis we refer to �ArBoMa� ��
��
We start the analysis by de�ning
���x� �% ��t�� �� exp��A�x� �% ��t��� � ���
and rewrite � ��a� in the usual way
�t � Ddivx
���rx�
�
���
�� � ���
We set up the relative entropy�type functional
e��j��� ��ZIRM�
ZIRnx
� ln��
���dxdP ���� � ���
Note that� strictly applying De�nition ���� e��j��� is not a relative entropy since�� is not necessarily normalized to � in L
��dxdP ����� Actually� the normalizationof �� was chosen such that
d
dte��j��� � I��j��� � � �
with the entropy production
I��j��� � �DZIRM�
ZIRnx
���jrx�
�
���j����dx�dP ��� � � ���
�cf� �ArBoMa� ��� We now proceed as in the linear case in Section � and computethe entropy production rate
d
dtI��j��� � ��I��j��� �D�
ZIRM
ZIRn
�MX
l�m�
��zl�xm
��
dxdP � ���
�D�
�ZIRM
ZIRn�jzj�dxdP � j
ZIRM
ZIRn�zdxdP j�
� �D
ZIRM
ZIRn
�z����W
�x�� I
�zdxdP
��DMX
l�m�
Elm
ZIRM
ZIRn
�l�zdxdP �ZIRM
ZIRn�m�zdxdP�
where z �� rx ln������ We remark that this computation follows the lines of the
proof of Lemma ���� where� in addition� the terms which come from the timedependence of ��� have to be taken care of �cf� �ArBoMa� � for details�� Todetermine in � ��� appropriately we assume that W is uniformly convex
�C�� �a � ��W�x��x� � aI x � IRn�
��
Since ZIRM
ZIRn
�l�zdxdP
� � ZIRM
��l dP ���
ZIRM
ZIRn
�jzj�dxdP
we can bound the last term on the right hand side of � ��� �from below� by
��De��ZIRM
ZIRn
�jzj�dxdP
assuming on the second moment of P�
�C��RIRM
j�j�dP ��� � � ��
Altogether we have
d
dtI��j��� � ��I��j��� f�t� � ���
where f�t� � if exists such that
�C�� a� e�� � �
Clearly� this is the case if either e� � �i� e� E is negative semi�de�nite� or� ife� � then the product e�� has to be smaller than the convexity bound a�
Assume now that satis�es �C��� Then � ��� implies
jI��j����t�j � e���tjI��I j���t � ��j� � ���
Since
,z��t� � �DZIRM
ZIRn�zdxdP� ,s��m�t� � �D
ZIRM
ZIRn
�m�zdxdP
we estimate �using I��j��� � �DRIRM
RIRn �jzj�dxdP ��
j ,z��t�j � e���tjI��I j���t � ��j� � ��a�
j ,s��m�t�j � �e���tjI��I j���t � ��j� � ��b�
Now we normalize ���
���x� �% ��t�� �����x� �% ��t��R
IRn���y� �% ��t��dy
ZIRn
�I�y� ��dy � ����
assuming
�
�C �RIRM e j�j
�dP ��� � for some � su!ciently large�
Obviously we have
I��j��� � I��j����Since
RIRnx��x� �� t�dx is conserved by the equation � ��a� we can apply the loga�
rithmic Sobolev�inequality ����� �with ���� � � ln��� � and a��Das convexity
constant�� ZIRn� ln
��
��
�dx � D
��a ��
ZIRn
�
�jr� ����j�dx�
After integration with respect to dP ���� �I��j��� appears on the right hand sideand the relative entropy of ��t� with respect to ���t� on the left hand side� Thus
e���t�j���t�� � De���t
��a ��jI��I j���t � ��j � ����
follows and the Csisz�ar�Kullback inequality gives�
k��t�� ���t�kL��dxdP ���� � ��DjI��I j���t � ��j
��a ��
� ��
e��t� � ����
The estimates � ��� imply that z���� �� limt�� z��t� and s��m��� �� limt�� s��m�t�exist and
jz��t�� z����j � �
�jI��I j���t � ��je���t� � ���a�
js��m�t�� s��m���j � ��jI��I j���t � ��je���t� � ���b�
Using these estimates we can �eliminate� the t�dependence of ���x� �% ��t�� asym�ptotically and prove
Theorem ���� Let �C���C�� hold and assume jI��Ij���t � ��j �� Then asteady state '�� � L���dxdP ���� of ����� exists and there is a constant K only depending on �I and on the parameters of the problem ����� such that
k��t�� '��kL��dxdP ���� � Ke��t t �
Clearly� '�� is a solution of the equation
'���x� �� ����x� �% '���R
IRn���y� �% '���dy
ZIRn
�I�y� ��dy� � �� �
��
If W �x� � W ��x� x � IRn holds� then a solution of � �� � can be easily con�structed� We set z� � s��m � and
'���x� �� � N exp�� �D�W �x�
�
�jxj��
�ZIRn
�I�y� ��dy� � ����
where N ��R
IRn exp�� �
D�W �y� �
�jyj��� dy���� It is immediately veri�ed that
this function solves the equation � �� ��
We remark that convergence of ��t� to a steady state '�� without a convergencerate can still be shown if the single oscillator potential W �x� is not uniformlyconvex but satis�es only certain mild growth conditions �cf� �ArBoMa� ��� Then
it was shown in �ArBoMa� � that I�t�t���� still holds �without a rate�� The
method used above implies immediately
e���t�j��� t����
and ��t�t���� '�� in L��dxdP ���� follows�
��� The drift�di usion�Poisson model
Secondly� we consider the drift�di�usion model with Poisson�coupling for an elec�tron gas �Mar���� �MaRiSc��
�t � div
�r� r
� jxj�� V
��
�� x � IRn� t � � ���a�
��t � � � �I � on IRn�
ZIRn
�I�x�dx � �� � ���b�
�&V � �� � ���c�
Here jxj��acts as a con�ning potential� For the sake of simplicity we only consider
the case of at least � dimensions� i�e� n � � and take the Newtonian solution of� ���c��
V �x� t� ��
�n� ��Sn
ZIRn
��y� t�
jx� yjn��dy� � ���d�
with Sn being the surface area of the unit sphere in IRn� An existence�uniqueness
result �for a global solution� of � ���� is easily obtained by proceeding in analogy
��
to the bounded domain case �cf� e�g� �Gaj� ��� The steady state of � ���� is theunique solution of the mean��eld equation
���x� �� exp��jxj
�
�� V��x�
��ZIRn
exp
��jyj
�
�� V��y�
�dy� � ���a�
V��x� ��
�n� ��Sn
ZIRn
���y�jx� yjn��dy� � ���b�
A detailed analysis of equations of the type � ���� can be found in �Dol����
We shall now show that ��t� converges exponentially to �� by using logarithmicSobolev inequalities� We start by de�ning the relative entropy type functional
e ��
ZIRn
� ln
��
��
�dx
�
�
ZIRn
jr�V � V��j�dx � ����
�cf� �Gaj� �� and compute its time�derivative�
d
dte�t� � �
ZIRn��t�jr ln
���t�
N�t�
�j�dx � ���
where we denoted the t�local state
N�t� � exp
��jxj
�
�� V �x� t�
��ZIRn
exp
��jyj
�
�� V �y� t�
�dy� � ����
We remark that the calculation which leads to � ��� is completely analogous to�Gaj� � making appropriate use of the Poisson equations � ���d�� � ���b��
Assume now that V �t� is in L��IRn� uniformly in t� i�e� there is a K suchthat
kV �t�kL��IRn� � K� t � � ����
�which shall be proven later on under additional assumptions on �I�� Then thelogarithmic Sobolev inequality ����� and the perturbation result of Theorem ���imply the existence of � �independent of t� such thatZ
� ln �N
�dx � �
��
Z�jr ln� �
N�j�dx� t � � � ����
We use � ���� in � ��� and obtain
d
dte�t� � ���
ZIRn
� ln �N
�dx � ���
ZIRn
� ln
��
��
�dx� ��
ZIRn
� ln��N
�dx�
��
Since
ln
���N�t�
�� V �t�� V� ln
�ZIRn
eV��V �t���dx�
we have
d
dte�t� � ���
ZIRn
� ln
��
��
�dx� ��
ZIRn
�V �t�� V���dx ��� ln�
ZIRn
eV��V �t���dx��
�usingRIRn��y� t�dy � � t �� The convexity of f�u� � � lnu implies
� ln�Z
IRn
eV��V �t���dx��ZIRn
�V �t�� V����dx
and
d
dte�t� � ���
ZIRn
� ln
��
��
�dx� ��
ZIRn�V �t�� V�����t�� ���dx
follows� Now we use
��t�� �� � �&�V �t�� V��such that
d
dte�t� � ���
ZIRn
��t� ln
���t�
��
�dx� ��
Zjr�V �t�� V��j�dx
� ���e�t��We conclude exponential convergence in relative entropy�Z
IRn
��t� ln
���t�
��
�dx
�
�
ZIRn
jr�V �t�� V��j�dx
� e����t�Z
IRn
�I ln
��I��
�dx
�
�
ZIRn
jr�VI � V��j�dx�
� ����
�where� obviously� VI is the Newtonian solution of �&VI � �I� and exponentialL��IRn��convergence of ��t� to �� follows from the Csisz�ar�Kullback inequality�
We are left with proving the bound � ����� Therefore we proceed as in �FaStr�and multiply � ���a� by �q� q � IN� Integration by parts and using � ���c� gives�
q �
ZIRn
�q��dx � � �q
q �
ZIRn
jr�� q��� �j�dx qn
q �
ZIRn
�q��dx� q
q �
ZIRn
�q��dx�
Applying the Nash�inequality �Nas ���ZIRn
f �dx
��� �n
� an�Z
IRn
jf jdx� �
nZIRn
jrf j�dx
��
for some an to the �rst term on the right hand side �with f � �q��� � gives
d
dtzq�� � ��q
an
�zq����� �
n
�z q�����n
nqzq���
where we set
zq ��
ZIRn
�qdx�
It is easy to conclude that zq���t� is uniformly bounded for t if zq���� ��By interpolation we �nd ��t� � Lp��IRn� uniformly for t if �I � Lp��IRn� L���IR
n��
Since the Newtonian potential of a function in Lp�IRn� is in L��IRn� if p n�we
obtain
Theorem ���� Let �I � L���IRn� Lp��IRn� for some p n�� Then there is
a constant � such that the exponential convergence ������ of the relativeentropy holds for the solution ���t�� V �t�� of �������
We refer to �ArMaTo��� for a detailed analysis of bipolar models of the form� ���� �i�e� with two types of carriers��
Appendix Proofs of Section �
Proof of Theorem ����
Step �� We shall �rst construct an auxiliary function �� We introduce
M �� f��� a� �� � J� � ����� �� �� � a minf��� � ��� ��� ���� � ��gg�We note that M � � consists exactly of those ��� a� �� � J� � ����� �� �� forwhich � �a��� and � � �a���� ��� belongs to J�� We de�ne� �M � IR
��� a� �� �� ��� a
�
�� ����
� ��� ��
��
�� � a
�� ��� ����
��
For �xed � � J� let Pra�M % �� be the projection of M onto the a�axes and for�xed � � J� and �xed a � Pra�M % �� let Pr��M % �� a� be the projection of Monto the ��axis� Then we can easily verify that
Pra�M % �� �
��Usup���
�
�� Pr��M % �� a� �
�a
� � � � ��a
� � ���
�
Due to the assumed strict convexity of � we observe that for all ��� a� � J� �� Usup��
�
��
inf��Pr��M ��a�
���� a� ��
� inf��Pr��M ��a�
a
���� a
�
�� ����a�
��� � a
����� ����
a���
�
Step �� We de�ne
-H � J� �� Usup��
�
�� ����
��� a� �� inf��Pr��M ��a�
���� a� ��
We note that -H is de�ned as the in�mum of a bounded�below� continuous function�recall that each convex function � is continuous on the interior of its domain�over an open set where two entries are �xed� Hence -H is continuous� Furthermore�by the strict convexity of �� the function
-H��� �� �� Usup��
�
�� ����
a �� -H��� a�
is for all �xed � � J� strictly increasing� It is not very di!cult to check that forall � � J��
lima��
-H��� a� � � lima�Usup���
�
-H��� a� � c�����
For �xed � � J� we can therefore de�ne the generalized inverse mapping �see�BeLo����
I��� �� � ���� ���Usup���
�
�
c ��
�����-H��� ��
����c� if c � �� c�����
Usup���
�if c � �c������
We note that I��� �� is increasing� strictly increasing on �� c����� and that themapping ��� c� �� I�c� ��� � � J�� c � ���� is continuous� Furthermore it iseasy to check that limc�� I�c� �� � � Let us note that
limc�c���
I�c� �� �Usup���
��
��
Step �� Now we are in the position to de�ne U�
U � ����� J � IR
�c� �� ������I�c� �� if c � �� c������ � � J�Usup��� if c � �c�������� � � J� if � � �J J
The veri�cation of P�� �P�� follows from the previous remarks and can thereforebe left to the reader�
Proof of Lemma �����
a�� b� co���� ��� is the envelope of a family of a!ne functions� Hence co���� ���is convex and continuous� see �EkTe����
d� � � and ��� � implies co���� ��� � � We easily deduce from B��$that �������� � ������ for all � � J� � � For each � � J� � J��� � there exists a � � ���� � �� �� such that �� � J� J�� We setK �� minf����������� ����������g� Then it is easy to verify that for alls� � J� and all s� � J��
���s�� � K �s� � ��� � ���s�� � K �s� � ����Hence� by de�nition� co���� ������ � K �� � ��� � ��� �� K � �
c� Let ��� �� � J� � J� with �� ��� We have to prove co���� ������� co���� �������� If �� � � this estimate will follow from d�� If �� � we set � ������������ Certainly � � �� ��� We calculate co���� ������� � co���� ����� �� � �� ��� � �co���� ����� �� � ��co���� ������� co���� �������� sinceco���� ����� � � �� � � �� �� and co���� ������� �
Proof of Theorem �����
a� After the previous remarks it su!ces to prove� For all f � L��d�� satisfyingB��and e��f jg� �� the pair �k�k� e����� �� �kf � gkL��d��� e��f jg�� � co�epi���� �epi������ Since both epi���� and epi���� are convex �subsets of IR�� we have
co�epi���� � epi�����
������ ��� ����� z� � �� � J�� �� � J�� � � �� ���
z � ������� ��� ��������� �Let )�
� �� ff � gg and )� �� ff gg� We de�ne � ��R���g d�� Then � � �� ��
and �� � � R��g d�� We introduce w �� �f�g�� � and set a �� R
���w g d�� as
well as b �� � R��w g d�� Then a� b � ���� and
k�k � a b�
��
We assume for the time being � � � �� Then
e���� � �
Z���
��� w�g
�d� ��� ��
Z����� w�
g
�� � d�
� ����
�
Z���
�g wg� d�
� ��� ���
��
�� �Z���g wg� d�
�� ��
� a
�
� ��� ���
��� b
�� ���
Setting
�� ��a
�� �� ��
b
�� ��
we obtain
k�k � ��� ��� ���� � e���� � ������� ��� ��������� ���� �
where �� � J� and �� � J�� The reader may wish to verify ���� � in caseof � � or � � � for some �� � J� and �� � J�� Hence �k�k� e����� �co�epi���� � epi������
b� For all � � J� � J� we have
co���� ������
� inff������� ��� �������� � �� � J�� �� � J�� � � �� ������ ��� ���� � �g�
Hence it is possible to choose a sequence ��n� ��n � �
�n �n�IN in �� ���J��J� with
�n��n ��� �n���n� � � for all n � IN �
limn��
�n�����n � ��� �n������n � � co���� �������
By assumption we can choose for all n � IN a set Sn � S with RSng d� � �n�
Setting for n � INfn � ) � IR
x ����� ��n � g�x� � x � Sn��� ��n � g�x� � x � ) n Sn
�nishes the proof�
��
References
�ArBoMa� � A� Arnold� L� Bonilla� P� Markowich� Liapunov functionals andlarge�time asymptotics of mean��eld Fokker�Planck equations�Transp� Theo� Stat� Phys� � ��������� �� �����
�ArMaTo��� A� Arnold� P� Markowich� G� Toscani� On large time asymptoticsfor drift�di�usion�Poisson systems� in preparation� �����
�BaEm��� D� Bakry� M� Emery� Hypercontractivit�e de semi�groupes de di�u�sion� C�R� Acad� Sc� Paris t� ��� S�erie I � � �����
�BaEm��� D� Bakry� M� Emery� Propaganda for "�� in From local times toglobal geometry control and physics� Pitman Res� Notes Math� Ser����� Longman Sci� Tech�� Harlow� ������ ����
�Bak��� D� Bakry� In�egalit�e de Sobolev faibles� un crit�ere "�� Lecture No�tes in Mathematics ���� J� Az�ema P�A� Meyer M� Yor �Eds���S�eminaire de Probabilit�es XXV� Springer Berlin� �����
�Bak��� D� Bakry� L�hypercontractivit�e et son utilisation en th�eoriedes semigroupes� Lecture Notes in Mathematics ���� D� Ba�kry R�D� Gill S�A� Molchanov �Eds�� �Lectures on ProbabilityTheory� Springer Berlin�Heidelberg� �����
�BaNi��� O� Barndor��Nielsen� Sub�elds and loss of information� Z�Wahrsch� Verw� Geb�� ���������� �����
�Bec��� W� Beckner� A generalized Poincar�e inequality for Gaussian mea�sures� Proc� Amer� Math� Soc�� � � ������ �����
�BeGo��� M� Berger� B� Gostiaux � Geometrie Di�erentielle� Varietes� Cour�bes et Surfaces� Presses Universitaires de France� �����
�BeLo��� J� Bergh� J� Lofstrom� Interpolation Spaces� Springer� �����
�BhGrKr �� P�L� Bhatnagar� E�P� Gross� M� Krook� A model for collision pro�cesses in gases� I� Small amplitude processes in charged and neutralone�component systems� Phys� Rev�� ��� ��� � � �� ��
�Bla� � N�M� Blachman� The convolution inequality for entropy powers�IEEE Trans� Info� Thy� ���������� ��� �
�Car��� E� Carlen� Superadditivity of Fisher�s Information and LogarithmicSobolev Inequalities� J� Func� Anal�� ����������� �����
��
�CeIlPu��� C� Cercignani� R� Illner� M� Pulvirenti The Mathematical Theoryof Dilute Gases SpringerVerlag� Berlin� ����
�Che��� H� Cherno�� A note on an inequality involving the normal distri�bution� Ann Probab� �� ��� � � �����
�Csi��� I� Csisz�ar� Eine informationstheoretische Ungleichung und ihre An�wendung auf den Beweis von Marko�schen Ketten� Magyar Tud�Akad� Mat� Kutat�o Int� K�ozl�� ��� ���� �����
�Csi��� I� Csisz�ar� A note on Jensen�s inequality� Stud� Sc� Math� Hung����������� �����
�Csi��� I� Csisz�ar� Information�type measures of di�erence of probabilitydistributions� Stud� Sc� Math� Hung�� ���������� �����
�Dav��� E� B� Davies� Heat Kernels and Spectral Theory� Cambridge Uni�versity Press� Cambridge� �����
�DeSt��� J� D� Deuschel� D� W� Stroock� Large Deviations� Academic Press�Boston� �����
�DeSt�� J� D� Deuschel� D� W� Stroock� Hypercontractivity and SpectralGap of Symmetric Di�usions with Applications to the StochasticIsing Model� J� Funct� Anal�� �������� ����
�DiSC��� P� Diaconis� L� Salo��Coste� Logarithmic Sobolev inequalities for�nite Markov chains� Ann� Appl� Probab�� ������� �� � �����
�Dol��� J� Dolbeault� Stationary states in plasma physics� Maxwellian so�lutions of the Vlasov�Poisson system� M�AS� ������������ �����
�EkTe��� I� Ekeland� R� Temam� Analyse convexe et probl�emes variationnelsDunod Gauthier�Villars� �����
�FaStr� E�B� Fabes� D�W� Stroock� A new Proof of Moser�s Parabolic Har�neck Inequality using the old Idea of Nash� Arch� Rat� Mech� Anal��Vol� ��� �������� �����
�Gaj� � H� Gajewski� On Existence� Uniqueness� and Asymptotic Beha�viour of Solutions of the Basic Equations for Carrier Transport inSemiconductors� ZAMM� � ���������� ��� �
�GaMaUn��� E� Gabetta� P� Markowich� A� Unterreiter� A note on the entropyproduction of the radiative transfer equation Preprint� �����
�
�Ger��� P� Gerard� Solutions Globales du Probl�eme de Cauchy pourl��equation de Boltzmann� Seminaire Bourbaki ���eme ann�ee� ������������
�Gro� � L� Gross� Logarithmic Sobolev inequalities� Amer� J� of Math������������� ��� �
�Gro�� L� Gross� Notes on the �Holley�Stroock perturbation lemma�� pri�vate communication� ����
�Gro��� L� Gross� Logarithmic Sobolev Inequalities and Contractivity Pro�perties of Semigroups� Lecture Notes in Mathematics �� � E� Fabeset al� �Eds�� �Dirichlet Forms� Springer Berlin�Heidelberg� �����
�HolSto��� R� Holley and D� Stroock� Logarithmic Sobolev Inequalities andStochastic Ising Models� J� Stat� Phys� � � /����� ������� �����
�Kul �� S� Kullback� Information Theory and Statistics� John Wiley� �� ��
�KuLei �� S� Kullback and R� Leibler� On information and su!ciency�Ann� Math� Stat�� ��������� �� ��
�Led��� M� Ledoux� On an Integral Criterion and Hypercontractivity ofDi�usion Semigroups and Extremal Functions� J� Funct� Anal��� ������� � �����
�Mar��� P�A� Markowich� The Stationary Semiconductor Device Equations�Springer New York� Wien� �����
�MaRiSc�� P�A� Markowich� C�A� Ringhofer� C� Schmeiser� SemiconductorEquations� Springer� New York� ����
�Nas �� J� Nash� Continuity of solutions of parabolic and elliptic equations�Amer� J� Math� ������� �� �� ��
�Per �� A� P�erez� Notions g�en�eralis�ees d�intertitude� d�entropie etd�information du point de vue de la th�eorie de martingales� Tran�sactions of the First Prague Conference on Information TheoryStatistical Decision Functions Random Processes Prague� ������� �� ��
�Per� � A� P�erez� Information� ��su!ciency and data reduction problems�Kybernetika� ���������� ��� �
�ReSi��� M� Reed� B� Simon� Methods of Modern Mathematical Physics II�Fourier Analysis� Self�Adjointness� Academic Press� �����
��
�ReSi��a� M� Reed� B� Simon� Methods of Modern Mathematical Physics IV�Analysis of Operators� Academic Press� �����
�Ris��� H� Risken� The Fokker�Planck Equation� Springer Verlag� �����
�Rot��� O� S� Rothaus� Di�usion on compact Riemannian manifolds andlogarithmic Sobolev inequalities� J� Funct� Anal�� ��������� �����
�Rot��a� O� S� Rothaus� Logarithmic Sobolev inequalities and the spectrumof Schrodinger operators� J� Funct� Anal�� ��������� �����
�Rud��� W� Rudin� Real and Complex Analysis� Mc�Graw Hill� �����
�Sim��� B� Simon� A remark on Nelson�s best hypercontractive estimate�Proc� Amer� Math� Soc�� ��������� �����
�Sta �� A�J� Stam� Some inequalities satis�ed by the quantities of informa�tion of Fisher and Shannon� Info� Contr�� ��������� �� ��
�Tos��a� G� Toscani� Kinetic approach to the asymptotic behaviour of thesolution to di�usion equations� Rend� di Matematica� ����������������
�Tos��b� G� Toscani� Entropy production and the rate of convergence toequilibrium for the Fokker�Planck equation� Quart� Appl� Math�to appear� �����
�Tos��a� G� Toscani� Sur l�in�egalit�e logarithmique de Sobolev� C�R� Acad�Sc� Paris� ���� �������� �����
�Tos��b� G� Toscani� The grazing collisions asymptotics of the non cut�o�Kac equation� M�AN to appear� �����
�Vil��� C� Villani� On the Landau equation � weak stability� global exi�stence� Adv� Di�� Eq�� �� �� �������� �����
�Vil��� C� Villani� On a new class of weak solutions to the spatially homo�geneous Boltzmann and Landau equations� Arch� Rat� Mech� Anal�������
�Wey��� H� Weyl� The Theory of Groups and Quantum Mechanics� DoverPublications� New York� �����
��