Effective Approximation for a Nonlocal Stochastic Schrödinger Equation with Oscillating Potential

Abstract

We study the effective approximation for a nonlocal stochastic Schrodinger equation with a rapidly oscillating, periodically time-dependent potential. We use the natural diffusive scaling of heterogeneous system and study the limit behaviour as the scaling parameter tends to 0. This is motivated by data assimilations with non-Gaussian uncertainties. The nonlocal operator in this stochastic partial differential equation is the generator of a non-Gaussian Levy-type process (i.e., a class of anomalous diffusion processes), with non-integrable jump kernel. With 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 Schrodinger equation has a nonlocal effective equation. We show that it approximates the orginal stochastic Schrödinger equation weakly in a Sobolev-type space and strongly in $L^2$ space. In particular, this effective approximattion holds when the nonlocal operator is the fractional Laplacian.