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

TL;DR

This paper analyzes the convergence of large deviations rate functions for numerical discretizations of stochastic wave equations with small noise, introducing a new technical approach based on $$-convergence.

Contribution

It develops a novel method for analyzing the pointwise convergence of large deviations rate functions using $$-convergence, addressing challenges in nonlinear stochastic wave equations.

Findings

Established convergence of one-point LDRFs for spatial FDM

Proposed a new technical route based on $$-convergence

Reduced analysis to qualitative study of skeleton equations

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.