Stochastic Homogenization of Reflected Diffusion Processes

TL;DR

This paper establishes a limit theorem for reflected stochastic differential equations with stationary coefficients, showing convergence to a reflected non-standard Brownian motion, and addresses challenges in non-translation invariant settings.

Contribution

It introduces a novel approach to homogenization for reflected diffusions in non-translation invariant environments, overcoming limitations of traditional methods.

Findings

Limiting process is a reflected non-standard Brownian motion.

Proves a functional limit theorem for reflected SDEs.

Addresses challenges in non-translation invariant homogenization.

Abstract

We investigate a functional limit theorem (homogenization) for Reflected Stochastic Differential Equations on a half-plane with stationary coefficients when it is necessary to analyze both the effective Brownian motion and the effective local time. We prove that the limiting process is a reflected non-standard Brownian motion. Beyond the result, this problem is known as a prototype of non-translation invariant problem making the usual method of the "environment as seen from the particle" inefficient.