Homogenization of a singular random one dimensional parabolic PDE with time varying coefficients

TL;DR

This paper investigates the homogenization of a one-dimensional parabolic PDE with rapidly oscillating random potentials, revealing conditions under which the limit is deterministic or stochastic, and establishing convergence types.

Contribution

It provides a rigorous analysis of homogenization for non-autonomous PDEs with random potentials, distinguishing between cases with temporal and spatial microstructures.

Findings

Deterministic limit when potential is homogeneous in both variables.

Stochastic limit when microstructure exists only in one variable.

Convergence in probability or law depending on potential structure.

Abstract

The paper studies homogenization problem for a non-autonomous parabolic equation with a large random rapidly oscillating potential in the case of one dimensional spatial variable. We show that if the potential is a statistically homogeneous rapidly oscillating function of both temporal and spatial variables then, under proper mixing assumptions, the limit equation is deterministic and the convergence in probability holds. To the contrary, for the potential having a microstructure only in one of these variables, the limit problem is stochastic and we only prove the convergence in law.