Large deviations principles of sample paths and invariant measures of numerical methods for parabolic SPDEs

TL;DR

This paper establishes that certain numerical methods for parabolic SPDEs satisfy uniform sample path large deviations and preserve the large deviations principles of invariant measures, ensuring accurate long-term probabilistic behavior approximation.

Contribution

It introduces a novel approach to proving weak asymptotic preservation of large deviations principles for both sample paths and invariant measures using numerical methods for SPDEs.

Findings

Numerical methods satisfy uniform sample path large deviations.

Invariant measures of numerical methods obey large deviations principles.

Numerical methods weakly asymptotically preserve the LDPs of the original SPDEs.

Abstract

For parabolic stochastic partial differential equations (SPDEs), we show that the numerical methods, including the spatial spectral Galerkin method and further the full discretization via the temporal accelerated exponential Euler method, satisfy the uniform sample path large deviations. Combining the exponential tail estimate of invariant measures, we establish the large deviations principles (LDPs) of invariant measures of these numerical methods. Based on the error estimate between the rate function of the considered numerical methods and that of the original equation, we prove that these numerical methods can weakly asymptotically preserve the LDPs of sample paths and invariant measures of the original equation. This work provides an approach to proving the weakly asymptotical preservation for the above two LDPs for SPDEs with small noise via numerical methods, by means of the minimization sequences.