Explicit approximation for stochastic nonlinear Schrödinger equation

Abstract

In this paper, we study explicit approximations of stochastic nonlinear Schrödinger equations (SNLSEs). We first prove that the classical explicit numerical approximations are divergent for SNLSEs with polynomial nonlinearities. To enhance the stability, we propose a kind of explicit numerical approximations, and establish the regularity analysis and strong convergence rate of the proposed approximations for SNLSEs. There are two key ingredients in our approach. One ingredient is constructing a logarithmic auxiliary functional and exploiting the Bourgain space to prove new regularity estimates of SNLSEs. Another one is providing a dedicated error decomposition formula and presenting the tail estimates of underlying stochastic processes. In particular, our result answers the strong convergence problem of numerical approximation for 2D SNLSEs.