Equilibrium with coordinate dependent diffusion: Comparison of different stochastic processes

TL;DR

This paper explores the symmetry properties of Itô-processes with coordinate-dependent diffusion, providing a mapping to a universal Gaussian form and analyzing the resulting equilibrium distributions and entropy in such stochastic systems.

Contribution

It introduces a symmetry transformation for Itô-processes with coordinate-dependent diffusion and uses it to analyze equilibrium states and entropy in these systems.

Findings

Identified a symmetry of Itô-processes under local coordinate and time scaling.

Mapped any Itô-process to a universal additive Gaussian-noise form.

Derived the equilibrium distribution and entropy for Brownian particles with coordinate-dependent diffusion.

Abstract

We show that, simultaneous local scaling of coordinate and time keeping the velocity unaltered is a symmetry of an It\^o-process. Using this symmetry, any It\^o-process can be mapped to a universal additive Gaussian-noise form. We use this mapping to separate the canonical and micro-canonical part of stochastic dynamics of a Brownian particle undergoing coordinate dependent diffusion. We identify the equilibrium distribution of the system and associated entropy induced by coordinate dependence of diffusion. Equilibrium physics of such a Brownian particle in a heat-bath of constant temperature is that of an It\^o-process.