Random homogenisation of a highly oscillatory singular potential

TL;DR

This paper studies the homogenisation of the heat equation with a rapidly oscillating random potential, focusing on the case where spatial oscillations dominate temporal ones, and proves convergence to a deterministic limit.

Contribution

It extends previous results by analyzing the homogenisation under a different scaling regime where spatial oscillations are faster than temporal ones.

Findings

Convergence to a deterministic heat equation with constant potential.

Homogenisation results under non-diffusive scaling regimes.

Completion of previous homogenisation analyses for different oscillation scalings.

Abstract

In this article, we consider the problem of homogenising the linear heat equation perturbed by a rapidly oscillating random potential. We consider the situation where the space-time scaling of the potential's oscillations is \textit{not} given by the diffusion scaling that leaves the heat equation invariant. Instead, we treat the case where spatial oscillations are much faster than temporal oscillations. Under suitable scaling of the amplitude of the potential, we prove convergence to a deterministic heat equation with constant potential, thus completing the results previously obtained in \cite{MR2962093}.