# Ergodic properties of anomalous diffusion processes

## Abstract

In this paper we study ergodic properties of some classes of anomalous diffusion processes. Using the recently developed measure of dependence called the Correlation Cascade, we derive a generalization of the classical Khinchin theorem. This result allows us to determine ergodic properties of Levy-driven stochastic processes. Moreover, we analyze the asymptotic behavior of two different fractional Ornstein-Uhlenbeck processes, both originating from subdiffusive dynamics. We show that only one of them is ergodic. - Highlights: > We derive a generalization of the classical Khinchin ergodic theorem for the general class of Levy-driven processes. > We study ergodic properties of stable and tempered stable processes. > We verify ergodicity and mixing of two fractional Ornstein-Uhlenbeck processes, both originating from subdiffusive dynamics.

- Authors:

- Publication Date:

- OSTI Identifier:
- 21583333

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 326; Journal Issue: 9; Other Information: DOI: 10.1016/j.aop.2011.04.015; PII: S0003-4916(11)00081-9; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; CORRELATIONS; DIFFUSION; FOKKER-PLANCK EQUATION; STOCHASTIC PROCESSES; DIFFERENTIAL EQUATIONS; EQUATIONS; MATHEMATICAL SOLUTIONS; PARTIAL DIFFERENTIAL EQUATIONS

### Citation Formats

```
Magdziarz, Marcin, E-mail: marcin.magdziarz@pwr.wroc.pl, and Weron, Aleksander.
```*Ergodic properties of anomalous diffusion processes*. United States: N. p., 2011.
Web. doi:10.1016/j.aop.2011.04.015.

```
Magdziarz, Marcin, E-mail: marcin.magdziarz@pwr.wroc.pl, & Weron, Aleksander.
```*Ergodic properties of anomalous diffusion processes*. United States. doi:10.1016/j.aop.2011.04.015.

```
Magdziarz, Marcin, E-mail: marcin.magdziarz@pwr.wroc.pl, and Weron, Aleksander. Thu .
"Ergodic properties of anomalous diffusion processes". United States. doi:10.1016/j.aop.2011.04.015.
```

```
@article{osti_21583333,
```

title = {Ergodic properties of anomalous diffusion processes},

author = {Magdziarz, Marcin, E-mail: marcin.magdziarz@pwr.wroc.pl and Weron, Aleksander},

abstractNote = {In this paper we study ergodic properties of some classes of anomalous diffusion processes. Using the recently developed measure of dependence called the Correlation Cascade, we derive a generalization of the classical Khinchin theorem. This result allows us to determine ergodic properties of Levy-driven stochastic processes. Moreover, we analyze the asymptotic behavior of two different fractional Ornstein-Uhlenbeck processes, both originating from subdiffusive dynamics. We show that only one of them is ergodic. - Highlights: > We derive a generalization of the classical Khinchin ergodic theorem for the general class of Levy-driven processes. > We study ergodic properties of stable and tempered stable processes. > We verify ergodicity and mixing of two fractional Ornstein-Uhlenbeck processes, both originating from subdiffusive dynamics.},

doi = {10.1016/j.aop.2011.04.015},

journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = 9,

volume = 326,

place = {United States},

year = {2011},

month = {9}

}