Random perturbations of dynamical systems with reflecting boundary and corresponding PDE with a small parameter

TL;DR

This paper investigates the long-term behavior of a reflected diffusion process with small diffusion in a domain and applies the findings to analyze the asymptotic behavior of the associated parabolic PDE with Neumann boundary conditions.

Contribution

It provides new asymptotic results for reflected diffusions with small noise and connects these to the long-time behavior of related PDE solutions.

Findings

Asymptotic behavior characterized for small diffusion reflected processes

Long-term PDE solutions analyzed with Neumann boundary conditions

Results applicable to understanding boundary effects in stochastic systems

Abstract

We study the asymptotic behavior of a diffusion process with small diffusion in a domain $D$. This process is reflected at $\partial D$ with respect to a co-normal direction pointing inside $D$. Our asymptotic result is used to study the long time behavior of the solution of the corresponding parabolic PDE with Neumann boundary condition.