Infinite Ergodic Theory for Heterogeneous Diffusion Processes

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 ${D(x)}\sim |x-\tilde{x}|^{2-2/α}$ in the vicinity of a point $\tilde{x}$, 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 $D(x)$.