## Abstract

We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite ergodic theory. We concentrate on a spatially dependent diffusion coefficient that behaves as D(x)∼|x-x|2-2/α in the vicinity of a point x, where α can be either positive or negative. We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. For processes under investigation, the time averages of a wide class of observables are obtained using an ensemble average with respect to the nonnormalized density. A Langevin equation which involves multiplicative noise may take different interpretation, Itô, Stratonovich, or Hänggi-Klimontovich, so the existence of an infinite density and the density's shape are both related to the considered interpretation and the structure of D(x).

Original language | English |
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Article number | 042138 |

Journal | Physical Review E |

Volume | 99 |

Issue number | 4 |

DOIs | |

State | Published - 24 Apr 2019 |

### Bibliographical note

Publisher Copyright:© 2019 American Physical Society.

### Funding

The support of Israel Science Foundation's Grant No. 1898/17 is acknowledged. We thank Guenter Radons, Jakub Ślęzak, and Takuma Akimoto for the discussion and comments.

Funders | Funder number |
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Israel Science Foundation | 1898/17 |