Stochastic homogenization of nonlinear evolution equations with space-time nonlocality

Abstract

In this paper we consider the homogenization problem of nonlinear evolution equations with space-time non-locality, the problems are given by Beltritti and Rossi [JMAA, 2017, 455: 1470-1504]. When the integral kernel $J(x,t;y,s)$ is re-scaled in a suitable way and the oscillation coefficient $\nu (x,t;y,s)$ possesses periodic and stationary structure, we show that the solutions $u^{\epsilon }(x,t)$ to the perturbed equations converge to $u_0(x,t)$, the solution of corresponding local nonlinear parabolic equation as scale parameter $\epsilon \to 0^{+}$. Then for the nonlocal linear index $p=2$ we give the convergence rate such that $||u^{\epsilon }-u_0|{|}_{{}_{L^2({\mathbb{R}}^d\times (0,T))}}\le C\epsilon$. Furthermore, we obtain that the normalized difference $\frac{1}{\epsilon }[u^{\epsilon }(x,t)-u_0(x,t)]-\chi (\frac{x}{\epsilon },\frac{t}{{\epsilon }^2}){\nabla }_x{u}_0(x,t)$ converges to a solution of an SPDE with additive noise and constant coefficients. Finally, we give some numerical formats for solving non-local space-time homogenization.

24 pages, 1 figure