Physical interpretation of It\^o--distribution on the basis of local measurement of diffusion

TL;DR

This paper offers a physical interpretation of the Itô-distribution for Brownian particles with coordinate-dependent diffusion, emphasizing the role of local diffusivity fluctuations in deriving the Gibbs measure.

Contribution

It introduces a new perspective linking local diffusivity fluctuations to the Hamiltonian modification, explaining the Itô-distribution physically.

Findings

Local diffusivity fluctuations are crucial in thermal equilibrium.

Modification of the Hamiltonian leads to the Gibbs measure.

Provides a physical basis for the Itô-distribution in diffusion processes.

Abstract

In this paper we provide a physical interpretation of It\^o-process resulting in thermal equilibrium distribution of a Brownian particle experiencing coordinate dependent diffusion. Since the local quantities like diffusivity would go through large fluctuations in thermal equilibrium, one needs to take these fluctuation into account. We identify that the definition of local diffusivity is an essential ingredient that effectively modifies Hamiltonian of the system to result in a physically relevant Gibbs measure related to the It\^o-distribution.