Homogenization for Space-Time-Dependent KPP Reaction-Diffusion Equations and G-Equations

TL;DR

This paper establishes stochastic homogenization for space-time-dependent KPP reaction-diffusion equations and G-equations, demonstrating deterministic spreading speeds in random media with decaying temporal correlations.

Contribution

It provides the first proof of stochastic homogenization for these equations with space-time randomness and decaying correlations, extending previous deterministic results.

Findings

Homogenized equations exhibit deterministic, direction-dependent spreading speeds.

Results apply to reaction-diffusion and G-equations with random flame speeds.

Key tools include a non-autonomous subadditive theorem and virtual linearity principles.

Abstract

We prove stochastic homogenization for reaction-advection-diffusion equations with random space-time-dependent KPP reactions with temporal correlations that are decaying in an appropriate sense. We show that the limiting homogenized dynamic has the simple form of spreading with some deterministic direction-dependent speeds from the support of the initial datum. We obtain analogous results for G-equations with random flame speeds and incompressible background advections. Important ingredients in our proofs are a non-autonomous subadditive theorem and the principle of virtual linearity for KPP reactions from the companion papers [30, 35].