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
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 2 / 31
Basic concepts in ergodic theory
Y (t), t ∈ R, stationary stochastic process
system is in thermal equilibrium
classical ergodic theorems apply
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 3 / 31
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)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 3 / 31
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.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 4 / 31
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
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 4 / 31
Mixing
Definition
The stationary process Y (t) is mixing if
limt→∞
P(A ∩ St(B)) = P(A)P(B)
for all A,B ∈ B.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 5 / 31
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
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 5 / 31
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))|) < ∞.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 6 / 31
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)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 7 / 31
Ergodic properties of Gaussian processes
Y (t) – stationary Gaussian process
Denote by r(t) autocorrelation function of Y (t). Then
G. Maruyama, Mem. Fac. Sci. Kyushu Univ. (1949)U. Grenander, Ark. Mat. (1950)K. Ito, Proc. Imp. Acad. (1944)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 7 / 31
Ergodic properties of Gaussian processes
Y (t) – stationary Gaussian process
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
G. Maruyama, Mem. Fac. Sci. Kyushu Univ. (1949)U. Grenander, Ark. Mat. (1950)K. Ito, Proc. Imp. Acad. (1944)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 7 / 31
Ergodic properties of Gaussian processes
Y (t) – stationary Gaussian process
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 mixing if and only if its autocorrelation function satisfies
limt→∞r(t) = 0. (1)
G. Maruyama, Mem. Fac. Sci. Kyushu Univ. (1949)U. Grenander, Ark. Mat. (1950)K. Ito, Proc. Imp. Acad. (1944)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 7 / 31
Ergodic properties of α-stable processes (Levy flights)
Problem: How to verify ergodic properties of α-stable processes(Levy flights)?
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 8 / 31
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
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 8 / 31
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
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.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 8 / 31
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
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.
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
Levy autocorrelation function
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)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 11 / 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).
Theorem (Magdziarz, 2010)
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,
where ν0t is the Levy measure of (Y0,Yt).
G. Maruyama, Theory Probab. Appl. (1970)M. Magdziarz, Theory Probab. Appl. (2010)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 11 / 31
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)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 12 / 31
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.
M. Magdziarz, Stoch. Proc. Appl. (2009)M. Magdziarz, A. Weron, Ann. Phys. (2011)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 12 / 31
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.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 13 / 31
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.
The Levy autocorrelation function corresponding to Y1(t) satisfies
R(t) ∝ e−αλt
as t → ∞. Thus, Y1(t) is ergodic and mixing.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 13 / 31
Examples - α-stable Ornstein-Uhlenbeck process
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).
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 14 / 31
Examples - α-stable Ornstein-Uhlenbeck process
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)).
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 15 / 31
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.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 16 / 31
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.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 16 / 31
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}.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 17 / 31
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}.
For α = 2 it reduces to the fractional Brownian motion.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 17 / 31
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}.
For α = 2 it reduces to the fractional Brownian motion.
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.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 17 / 31
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}.
For α = 2 it reduces to the fractional Brownian motion.
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.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 17 / 31
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.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 18 / 31
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
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 18 / 31
Levy autocorrelation function
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.
M. Magdziarz, Stoch. Proc. Appl. (2009)M. Magdziarz, A. Weron, Ann. Phys. (2011)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 20 / 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.
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.
M. Magdziarz, Stoch. Proc. Appl. (2009)M. Magdziarz, A. Weron, Ann. Phys. (2011)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 20 / 31
Summary for Levy autocorrelation function
Levy autocorrelation function seems to be a perfect tool forverification of ergodic properties of Levy flights
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 21 / 31
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.)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 21 / 31
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.)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)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 21 / 31
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.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 22 / 31
Experiment. Bar Ilan University, prof. J. Garini group
The telomeres can be labeled in living cells using fluorescent proteins.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 23 / 31
Experiment
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 24 / 31
Telomeres: - ergodic and mixing
Figure: Sample trajectories of telomeres.
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 25 / 31
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).
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 26 / 31
Generalized diffusion equation (GDE)
Definition (I.M. Sokolov, J. Klafter, Phys. Rev. Lett. (2006))
∂w(x , t)
∂t= Φt
∂2
∂x2w(x , t)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 27 / 31
Generalized diffusion equation (GDE)
Definition (I.M. Sokolov, J. Klafter, Phys. Rev. Lett. (2006))
∂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
Generalized diffusion equation (GDE)
Definition (I.M. Sokolov, J. Klafter, Phys. Rev. Lett. (2006))
∂w(x , t)
∂t= Φt
∂2
∂x2w(x , t)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 28 / 31
Generalized diffusion equation (GDE)
Definition (I.M. Sokolov, J. Klafter, Phys. Rev. Lett. (2006))
∂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).
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 28 / 31
Generalized diffusion equation (GDE)
Definition (I.M. Sokolov, J. Klafter, Phys. Rev. Lett. (2006))
∂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)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 28 / 31
Generalized diffusion equation (GDE)
Definition (I.M. Sokolov, J. Klafter, Phys. Rev. Lett. (2006))
∂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)
T – any infinitely divisible distribution
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 28 / 31
Generalized diffusion equation (GDE)
Definition (I.M. Sokolov, J. Klafter, Phys. Rev. Lett. (2006))
∂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)
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)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 29 / 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).
Lemma
Let E(T ) < ∞. Then the increments of SΨ(t) are asymptoticallystationary.
M. Magdziarz (2010)M. Magdziarz, R.L. Schilling (2013)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 29 / 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).
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.
M. Magdziarz (2010)M. Magdziarz, R.L. Schilling (2013)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 29 / 31
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
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 30 / 31
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)
Marcin Magdziarz (Wrocław) Ergodicity and mixing of anomalous diff. Wrocław 2013 31 / 31