Periodic Homogenization for Switching Diffusions

TL;DR

This paper develops homogenization techniques for switching diffusion processes with periodic coefficients, revealing how switching impacts the effective long-term behavior and diffusivity of the system.

Contribution

It extends classical homogenization methods to switching diffusions, explicitly computing the additional contribution to diffusivity caused by switching.

Findings

Derived the effective Brownian motion limit for scaled switching diffusions.

Quantified the additional diffusivity contribution from switching.

Provided explicit formulas for the covariance matrix of the limiting process.

Abstract

In the present work, we explore homogenization techniques for a class of switching diffusion processes whose drift and diffusion coefficients, and jump intensities are smooth, spatially periodic functions; we assume full coupling between the continuous and discrete components of the state. Under the assumptions of uniform ellipticity of the diffusion matrices and irreducibility of the matrix of switching intensities, we explore the large-scale long-time behavior of the process under a diffusive scaling. Our main result characterizes the limiting fluctuations of the rescaled continuous component about a constant velocity drift by an effective Brownian motion with explicitly computable covariance matrix. In the process of extending classical periodic homogenization techniques for diffusions to the case of switching diffusions, our main quantitative finding is the computation of an extra contribution to the limiting diffusivity stemming from the switching.