Infinite Ergodic Theory for Heterogeneous Diffusion Processes

TL;DR

This paper explores the connection between Langevin equations with multiplicative noise and infinite ergodic theory, revealing how nonnormalized densities describe statistical properties of heterogeneous diffusion processes with spatially dependent diffusion coefficients.

Contribution

It establishes the relationship between multiplicative noise Langevin processes and infinite ergodic theory, highlighting how interpretation affects the existence and shape of infinite densities.

Findings

Infinite densities describe statistical properties of the system.

Time averages can be computed using ensemble averages with respect to the infinite density.

The interpretation of the Langevin equation influences the existence and form of the infinite density.

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 ${D(x)}\sim |x-\tilde{x}|^{2-2/\alpha}$ in the vicinity of a point $\tilde{x}$, where $\alpha$ 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; It\^o, Stratonovich, or H\"anggi-Klimontovich, so the existence of an infinite density, and the density's shape, are both related to the considered interpretation and the structure of $D(x)$.