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 $u^{\varepsilon}(x,t)$ to the perturbed equations converge to $u_{0}(x,t)$, the solution of corresponding local nonlinear parabolic equation as scale parameter $\varepsilon\rightarrow 0^{+}$. Then for the nonlocal linear index $p=2$ we give the convergence rate such that $||u^\varepsilon -u_{0}||_{_{L^{2}(\mathbb{R}^{d}\times(0,T))}}\leq C\varepsilon$. Furthermore, we obtain that the normalized difference $\frac{1}{\varepsilon}[u^{\varepsilon}(x,t)-u_{0}(x,t)]-\chi(\frac{x}{\varepsilon}, \frac{t}{\varepsilon^{2}}) \nabla_{x}u_{0}(x,t)$ 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.