Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix

TL;DR

This paper advances the homogenization theory for locally stationary diffusions by incorporating possibly degenerate diffusion matrices, revealing how particles can be confined within subspaces of the ambient space.

Contribution

It extends previous homogenization results to include degenerate diffusion matrices, providing a more general framework for locally stationary diffusions.

Findings

Homogenization accounts for microscopic and macroscopic scales.

Degenerate diffusion matrices can trap particles in subspaces.

The homogenized equation reflects the geometry of particle confinement.

Abstract

This paper deals with homogenization of second order divergence form parabolic operators with locally stationary coefficients. Roughly speaking, locally stationary coefficients have two evolution scales: both an almost constant microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims to give a macroscopic approximation that takes into account the microscopic heterogeneities. This paper follows "Diffusion in a locally stationary random environment" (published in Probability Theory and Related Fields) and improves this latter work by considering possibly degenerate diffusion matrices. The geometry of the homogenized equation shows that the particle is trapped in subspace of R^d.