Fundamental solution expansions in stochastic homogenization of parabolic systems with time-dependent coefficients

Abstract

This paper studies fundamental solutions of uniformly parabolic systems with stationary random coefficients depending on space and time. Under ergodicity and local regularity, we establish quenched Gaussian bounds for the kernel and its spatial derivatives, with stationary random prefactors. A spectral gap assumption gives moments of every finite order for these prefactors and hence annealed Gaussian bounds. We also prove quantitative two-scale expansions for the kernel, its spatial gradient, and its mixed spatial derivative. The error estimates hold in every finite probability moment and retain Gaussian decay. Their dimension-dependent growth is described by a corrector weight, with a single additional logarithmic factor in the derivative estimates. The proofs combine large-scale regularity, extended correctors, weighted estimates, and a parabolic weak-norm estimate. The latter supplies the passage from deterministic pairings to local annealed estimates without interchanging expectation and a supremum over test functions. The mixed-derivative estimate builds on the gradient expansion and does not introduce a further logarithmic loss.