Convergence analysis of one-point large deviations rate functions of numerical discretizations for stochastic wave equations with small noise

Abstract

In this work, we present the convergence analysis of one-point large deviations rate functions (LDRFs) of the spatial finite difference method (FDM) for stochastic wave equations with small noise, which is essentially about the asymptotical limit of minimization problems and not a trivial task for the nonlinear cases. In order to overcome the difficulty that objective functions for the original equation and the spatial FDM have different effective domains, we propose a new technical route for analyzing the pointwise convergence of the one-point LDRFs of the spatial FDM, based on the $\Gamma$-convergence of objective functions. Based on the new technical route, the intractable convergence analysis of one-point LDRFs boils down to the qualitative analysis of skeleton equations of the original equation and its numerical discretizations.