Limit theorems for sticky SDEs with local times and applications to stochastic homogenization

TL;DR

This paper proves a convergence theorem for multivariate SDEs with local times, showing how diffusions with sticky interfaces homogenize as interface distances shrink, affecting drift and diffusion properties.

Contribution

It introduces a general limit theorem for SDEs with local times, modeling diffusions with sticky, semipermeable interfaces and their homogenization behavior.

Findings

Local-time terms converge to a homogenized drift as interfaces become dense.

Sticky interfaces cause the limiting diffusion to slow down.

The results demonstrate stochastic homogenization in heterogeneous media.

Abstract

In this paper, we establish a general convergence theorem for solutions of multivariate stochastic differential equations with countably many singular terms expressed as integrals with respect to local times. The processes under consideration describe diffusions in the presence of semipermeable hyperplane interfaces. These interfaces may become sticky after applying a random time change that depends on the amount of local time accumulated on each interface. We show that, as the distance between the interfaces tends to zero, the local-time terms converge to a limiting homogenized drift term. When the interfaces are sticky, the limiting diffusion also decelerates, meaning that its diffusion coefficient is effectively reduced. Such limit theorems illustrate a form of stochastic homogenization for diffusions evolving in a heterogeneous medium interleaved with semipermeable, sticky interfaces.