Effective approximation for a nonlocal stochastic Schrödinger equation with oscillating potential

Abstract

We study the effective approximation for a nonlocal stochastic Schrödinger equation with a rapidly oscillating, periodically time-dependent potential. We use the natural diffusive scaling of heterogeneous system and study the limit behavior as the scaling parameter tends to 0. This is motivated by data assimilation with non-Gaussian uncertainties. The nonlocal operator in this stochastic partial differential equation is the generator of a non-Gaussian Lévy-type process (i.e., a class of anomalous diffusion processes), with non-integrable jump kernel. With the help of a two-scale convergence technique, we establish effective approximation for this nonlocal stochastic partial differential equation. More precisely, we show that a nonlocal stochastic Schrödinger equation has a nonlocal effective equation. We show that it approximates the original stochastic Schrödinger equation weakly in a Sobolev-type space and strongly in \(L^2\) space. In particular, this effective approximation holds when the nonlocal operator is the fractional Laplacian.