Stationary distributions for diffusions with inert drift

TL;DR

This paper studies a reflecting diffusion process with inert drift in a domain, showing its stationary distribution has a product form with a symmetrizing measure and a Gaussian component, extending understanding of such stochastic systems.

Contribution

It establishes the explicit form of the stationary distribution for diffusions with inert drift, including cases with potential-driven drift, advancing theoretical knowledge in stochastic processes.

Findings

Stationary distribution has a product form involving symmetrizing measure and Gaussian distribution.

The symmetrizing measure corresponds to the diffusion without inert drift.

Results extend to processes with gradient-based drift.

Abstract

Consider a reflecting diffusion in a domain in $R^d$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting process and the value of the drift vector has a product form. Moreover, the first component is the symmetrizing measure on the domain for the reflecting diffusion without inert drift, and the second component has a Gaussian distribution. We also consider processes where the drift is given in terms of the gradient of a potential.