Quantitative homogenization for combustion in random media

Abstract

We obtain the first quantitative stochastic homogenization result for reaction–diffusion equations, for ignition reactions in dimensions $d\le 3$ that either have finite ranges of dependence or are close enough to such reactions, and for solutions with initial data that approximate characteristic functions of general convex sets. We show an algebraic rate of convergence of these solutions to their homogenized limits, which are (discontinuous) viscosity solutions of certain related Hamilton–Jacobi equations.