ergodicity and mixing of anomalous diffusion...

Ergodicity and mixing of anomalous diffusion processes

Marcin Magdziarz

Hugo Steinhaus Center

Institute of Mathematics and Computer Science

Wrocław University of Technology, Poland

7th International Conference on Levy Processes

Wrocław, July 2013

Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 1 / 31

Outline

Basic concepts in ergodic theory (ergodicity, mixing)Ergodic properties of α-stable processes (Levy flights)– Levy autocorrelation function– Refinement of Maruyama’s mixing theorem– Examples– Applications– Extension to infinitely divisible case

Ergodic properties of the generalized diffusion equation


Basic concepts in ergodic theory

Y (t), t ∈ R, stationary stochastic process

system is in thermal equilibrium

classical ergodic theorems apply


Basic concepts in ergodic theory

Y (t), t ∈ R, stationary stochastic process

system is in thermal equilibrium

classical ergodic theorems apply

Corresponding dynamical system (canonical representation of Y )

(RR,B,P,St)

where

RR – space of all functions f : R → R

B – Borel sets

P – probability measure

St – shift transformation, St(f )(s) = f (t + s)


Ergodicity

Definition

The stationary process Y (t) is ergodic if for every invariant set A we haveP(A) = 0 or P(Ac) = 0.

The set A is invariant if St(A) = A for all t.


Ergodicity

Definition

The stationary process Y (t) is ergodic if for every invariant set A we haveP(A) = 0 or P(Ac) = 0.

The set A is invariant if St(A) = A for all t.

Interpretation of ergodicity:

the space cannot be divided into two regions such that a pointstarting in one region will always stay in that region

the point will eventually visit all nontrivial regions of the space


Mixing

Definition

The stationary process Y (t) is mixing if

limt→∞

P(A ∩ St(B)) = P(A)P(B)

for all A,B ∈ B.


Mixing

Definition

The stationary process Y (t) is mixing if

limt→∞

P(A ∩ St(B)) = P(A)P(B)

for all A,B ∈ B.

Interpretation of mixing:

it can be viewed as an asymptotic independence of the sets A and Bunder the transformation Stthe fraction of points starting in A that ended up in B after long timet, is equal to the product of probabilities of A and B

Remark. Mixing is stronger property than ergodicity


Birkhoff ergodic theorem

Theorem

If the stationary process Y (t) is ergodic, then

limT→∞

1T

∫ T

0

g(Y (t))dt = E(g(Y (0))),

provided that E(|g(Y (0))|) < ∞.


Ergodic properties of Gaussian processes

Y (t) – stationary Gaussian process

G. Maruyama, Mem. Fac. Sci. Kyushu Univ. (1949)U. Grenander, Ark. Mat. (1950)K. Ito, Proc. Imp. Acad. (1944)




Denote by r(t) autocorrelation function of Y (t). Then






Theorem

Y (t) is ergodic if and only if its autocorrelation function satisfies

limT→∞1T

∫ T0r(t)dt = 0






Theorem

Y (t) is ergodic if and only if its autocorrelation function satisfies

limT→∞1T

∫ T0r(t)dt = 0

Theorem

Y (t) is mixing if and only if its autocorrelation function satisfies

limt→∞r(t) = 0. (1)



Ergodic properties of α-stable processes (Levy flights)

Problem: How to verify ergodic properties of α-stable processes(Levy flights)?




Main difficulty: the second moment is infinite – autocorrelationfunction is not defined




Main difficulty: the second moment is infinite – autocorrelationfunction is not defined

Examples of Levy flight dynamics: animal foraging patterns,transport of light in special optical materials, bulk mediated surfacediffusion, transport in micelle systems or heterogeneous rocks, singlemolecule spectroscopy, wait-and-switch relaxation, etc.


Levy autocorrelation function

Y (t) – stationary α-stable process (Levy flight) of the form

Y (t) =

∫ ∞

−∞K (t, x)dLα(x), t ∈ R. (2)

Here, K (t, x) is the kernel function and Lα(x) is the α-stable Levymotion with the Fourier transform Ee izLα(x) = e−x |z |

α

, 0 < α < 2.

I. Eliazar, J. Klafter, Physica A (2007); J. Phys. A: Math. Theor. (2007)Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 9 / 31


Y (t) – stationary α-stable process (Levy flight) of the form

Y (t) =

∫ ∞

−∞K (t, x)dLα(x), t ∈ R. (2)

Here, K (t, x) is the kernel function and Lα(x) is the α-stable Levymotion with the Fourier transform Ee izLα(x) = e−x |z |

α

, 0 < α < 2.

Definition (Levy autocorrelation function)

Levy autocorrelation function corresponding to Y (t) is defined as

R(t) =

∫ ∞

−∞min{|K (0, x)|, |K (t, x)|}αdx (3)

I. Eliazar, J. Klafter, Physica A (2007); J. Phys. A: Math. Theor. (2007)Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 9 / 31


Interpretation: For every l > 0 we have

R(t) = lα · ν0t{(x , y) : min{|x |, |y |} > l},

where ν0t is the Levy measure of the vector (Y (0),Y (t)).

l

OY

OXl−l

−l

Remark: Y (0) and Y (t) are independent if and only if ν0t is concentratedon the axes OX and OY.Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 10 / 31

Maruyama’s mixing theorem and its refinement

Theorem (Maruyama, 1970)

An infinitely divisible stationary process Yt is mixing if and only if

(i) autocorrelation r(t) of Gaussian part converges to 0 as t → ∞,

(ii) limt→∞ ν0t(|xy | > δ) = 0 for every δ > 0,

(iii) limt→∞

∫0<x2+y2≤1 xyν0t(dx , dy) = 0,

where ν0t is the Levy measure of (Y0,Yt).

G. Maruyama, Theory Probab. Appl. (1970)M. Magdziarz, Theory Probab. Appl. (2010)


Maruyama’s mixing theorem and its refinement

Theorem (Maruyama, 1970)




(iii) limt→∞

∫0<x2+y2≤1 xyν0t(dx , dy) = 0,


Theorem (Magdziarz, 2010)





G. Maruyama, Theory Probab. Appl. (1970)M. Magdziarz, Theory Probab. Appl. (2010)


Ergodic properties of Levy flights

Theorem

The stationary Levy flight process Y (t) is ergodic if and only if its Levyautocorrelation function satisfies

limT→∞

1T

∫ T

0

R(t)dt = 0.

M. Magdziarz, Stoch. Proc. Appl. (2009)M. Magdziarz, A. Weron, Ann. Phys. (2011)


Ergodic properties of Levy flights

Theorem

The stationary Levy flight process Y (t) is ergodic if and only if its Levyautocorrelation function satisfies

limT→∞

1T

∫ T

0

R(t)dt = 0.

Theorem

The stationary Levy flight process Y (t) is mixing if and only if its Levyautocorrelation function satisfies

limt→∞R(t) = 0.



Examples - α-stable Ornstein-Uhlenbeck process

α-stable Ornstein-Uhlenbeck process is defined as

Y1(t) = σ

∫ t

−∞e−λ(t−x)dLα(x), λ, σ > 0.



α-stable Ornstein-Uhlenbeck process is defined as

Y1(t) = σ

∫ t

−∞e−λ(t−x)dLα(x), λ, σ > 0.

The Levy autocorrelation function corresponding to Y1(t) satisfies

R(t) ∝ e−αλt

as t → ∞. Thus, Y1(t) is ergodic and mixing.



0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

t

R(0

,t)

α=0.3α=0.6α=0.9α=1.2α=1.5α=1.8

Figure: Levy autocorrelation function corresponding to the α-stableOrnstein-Uhlenbeck process Y1(t).



Figure: Top panel: simulated trajectory of the 1.2-stable Ornstein-Uhlenbeckprocess Y1(t). Bottom panel: the temporal average corresponding to Y1(t).

Clearly limT→∞

1T

∫ T0Y1(t)dt = 0 = E(Y1(0)).


Examples - α-stable Levy noise

α-stable Levy noise is defined as

lα(t) = Lα(t + 1)− Lα(t).

It is a stationary sequence of independent and identically distributedα-stable random variables.


Examples - α-stable Levy noise

α-stable Levy noise is defined as

lα(t) = Lα(t + 1)− Lα(t).

It is a stationary sequence of independent and identically distributedα-stable random variables.

The Levy autocorrelation function of lα(t) satisfies

R(t) = 0

This corresponds to the well known property that independent randomvariables are uncorrelated. Thus, lα(t) is ergodic and mixing.


Examples - fractional α-stable Levy motion

Fractional α-stable Levy motion is defined as

Lα,H(t) =

∫ ∞

−∞

[(t − x)

H−1/α+ − (−x)

H−1/α+

]dLα(x).

Here x+ = max{x , 0}.




Lα,H(t) =

∫ ∞

−∞

[(t − x)

H−1/α+ − (−x)

H−1/α+

]dLα(x).


For α = 2 it reduces to the fractional Brownian motion.




Lα,H(t) =

∫ ∞

−∞

[(t − x)

H−1/α+ − (−x)

H−1/α+

]dLα(x).



The stationary process of increments

lα,H(t) = Lα,H(t + 1)− Lα,H(t)

t ∈ N, is called the fractional α-stable Levy noise.




Lα,H(t) =

∫ ∞

−∞

[(t − x)

H−1/α+ − (−x)

H−1/α+

]dLα(x).



The stationary process of increments

lα,H(t) = Lα,H(t + 1)− Lα,H(t)

t ∈ N, is called the fractional α-stable Levy noise.

The Levy autocorrelation function of lα,H(t) yields

limt→∞R(t) = 0.

Therefore, the fractional α-stable Levy noise is ergodic and mixing.


Extension to infinitely divisible (i.d.) case

Y (t) – stationary i.d. process of the form

Y (t) =

∫ ∞

−∞K (t, x)dL(x), t ∈ R. (4)

Here, K (t, x) is the kernel function and L(x) is the Levy process withno Gaussian part and Levy measure Q.


Extension to infinitely divisible (i.d.) case

Y (t) – stationary i.d. process of the form

Y (t) =

∫ ∞

−∞K (t, x)dL(x), t ∈ R. (4)

Here, K (t, x) is the kernel function and L(x) is the Levy process withno Gaussian part and Levy measure Q.

Definition (Levy autocorrelation function)

Levy autocorrelation function corresponding to Y (t) is defined as

Rl(t) =1cl

∫ ∞

−∞Λ

(l

min{|K (0, x)|, |K (t, x)|}

)dx , l > 0. (5)

Here Λ(l) = Q(|x | > l) is the tail of the Levy measure Q andcl =

∫∞−∞ Λ( l

|K(0,x)|)dx



Interpretation: For every l > 0 we have

Rl(t) =1cl

· ν0t{(x , y) : min{|x |, |y |} > l},

where ν0t is the Levy measure of the vector (Y (0),Y (t)).

l

OY

OXl−l

−l

Remark: Y (0) and Y (t) are independent if and only if ν0t is concentratedon the axes OX and OY.Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 19 / 31

Ergodic properties of i.d. processes

Theorem

The stationary i.d. process Y (t) is ergodic if and only if its Levyautocorrelation function satisfies

limT→∞

1T

∫ T

0

Rl(t)dt = 0 for every l > 0.



Ergodic properties of i.d. processes

Theorem

The stationary i.d. process Y (t) is ergodic if and only if its Levyautocorrelation function satisfies

limT→∞

1T

∫ T

0

Rl(t)dt = 0 for every l > 0.

Theorem

The stationary i.d. process Y (t) is mixing if and only if its Levyautocorrelation function satisfies

limt→∞Rl(t) = 0 for every l > 0.



Summary for Levy autocorrelation function

Levy autocorrelation function seems to be a perfect tool forverification of ergodic properties of Levy flights




it works also for the whole family of infinitely divisible processes(α-stable, tempered α-stable, Pareto, exponential, gamma, Poisson,Linnik, Mittag-Leffler, etc.)




it works also for the whole family of infinitely divisible processes(α-stable, tempered α-stable, Pareto, exponential, gamma, Poisson,Linnik, Mittag-Leffler, etc.)Other approachesA. Weron et al. (1987, ...) - Dynamical functional

D(n) = Ee i [Y (n)−Y (0)]

J. Rosiński and T. Żak (1996, 1997) - CodifferenceE. Roy (2007)


Applications: Telomeres - ergodic and mixing

Each chromosome is capped in both ends with a telomere, that ismade of a repetitive DNA sequence and a set of proteins.

The telomeres prevent degradation of the ends of the chromosomesduring replication.

The 2009 Nobel Prize in Physiology or Medicine was given toElizabeth H. Blackburn, Carol W. Greider and Jack W. Szostak forthe discovery of how chromosomes are protected by telomeres and theenzyme telomerase.


Experiment. Bar Ilan University, prof. J. Garini group

The telomeres can be labeled in living cells using fluorescent proteins.


Experiment


Telomeres: - ergodic and mixing

Figure: Sample trajectories of telomeres.


Applications: Golding-Cox experiment - ergodic and mixing

0 500 1000 1500 2000−0.45

−0.3

−0.15

0

t

Y(t

)

Figure: Experimentally measured trajectory of fluorescently labeled mRNAmolecules inside live E. coli cells (Golding-Cox experiment, Princeton, 2006).


Generalized diffusion equation (GDE)

Definition (I.M. Sokolov, J. Klafter, Phys. Rev. Lett. (2006))

∂w(x , t)

∂t= Φt

∂2

∂x2w(x , t)




∂w(x , t)

∂t= Φt

∂2

∂x2w(x , t)

0 1 2 3 4−2

0

2

4

6

8

10

t

X(t

)

Figure: Typical trajectories of the process corresponding to GDEMarcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 27 / 31



∂w(x , t)

∂t= Φt

∂2

∂x2w(x , t)




∂w(x , t)

∂t= Φt

∂2

∂x2w(x , t)

Here

Φt f (t) =d

dt

∫ t

0

M(t − y)f (y)dy

and

M(u) =

∫ ∞

0

e−utM(t)dt =1

Ψ(u).




∂w(x , t)

∂t= Φt

∂2

∂x2w(x , t)

Here

Φt f (t) =d

dt

∫ t

0

M(t − y)f (y)dy

and

M(u) =

∫ ∞

0

e−utM(t)dt =1

Ψ(u).

Ψ(u) is the Laplace exponent of the underlying waiting time T > 0,i.e. E

(e−uT

)= e−Ψ(u)




∂w(x , t)

∂t= Φt

∂2

∂x2w(x , t)

Here

Φt f (t) =d

dt

∫ t

0

M(t − y)f (y)dy

and

M(u) =

∫ ∞

0

e−utM(t)dt =1

Ψ(u).


(e−uT

)= e−Ψ(u)

T – any infinitely divisible distribution




∂w(x , t)

∂t= Φt

∂2

∂x2w(x , t)

Here

Φt f (t) =d

dt

∫ t

0

M(t − y)f (y)dy

and

M(u) =

∫ ∞

0

e−utM(t)dt =1

Ψ(u).


(e−uT

)= e−Ψ(u)

T – any infinitely divisible distribution

for Ψ(u) = uα we have Φt = 0D1−αt and we recover the celebrated

fractional diffusion equationMarcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 28 / 31

GDE – ergodicity and mixing

Theorem

The PDF of the process X (t) = B(SΨ(t)) is the solution of GDE. Here, Bis the Brownian motion and SΨ is the inverse subordinator corresponding

to T , i.e. SΨ = inf{τ > 0 : TΨ(τ) > t} with Ee−uTΨ(t) = e−tΨ(u).

M. Magdziarz (2010)M. Magdziarz, R.L. Schilling (2013)



Theorem



Lemma

Let E(T ) < ∞. Then the increments of SΨ(t) are asymptoticallystationary.




Theorem



Lemma

Let E(T ) < ∞. Then the increments of SΨ(t) are asymptoticallystationary.

Theorem

Let E(T ) < ∞. Then the stationary increments of the processX (t) = B(SΨ(t)) corresponding to GDE are ergodic and mixing.



Applications: Lipid granules - ergodic and mixing

Recently, in J-H. Jeon et al., Phys. Rev. Lett (2010), GDE with temperedstable waiting times was used to model the dynamics of lipid granules infission yeast cells. The above theorem implies that this dynamics isergodic and mixing.

Figure: Trajectory of lipid granules motion


The end

References

A. Janicki and A. Weron, Simulation and Chaotic Behaviour ofα-Stable Stochastic Processes (Marcel Dekker, 1994)

M. Magdziarz, Stoch. Proc. Appl. (2009)

M. Magdziarz, Theory Probab. Appl. (2010)

M. Magdziarz, A. Weron, Ann. Phys. (2011)

K. Burnecki, E. Kepten, J. Janczura, I. Bronshtein, Y. Garini, A.Weron, Biophysical Journal (2012)

M. Magdziarz, R.L. Schilling, preprint (2013)


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