Weak convergence of finite element approximations of linear stochastic evolution equations with additive L\'evy noise

TL;DR

This paper develops a general framework to analyze the weak convergence of finite element methods for linear stochastic PDEs driven by additive Lévy noise, extending previous results to more general equations and noise types.

Contribution

It introduces a novel abstract approach for weak convergence analysis applicable to hyperbolic, parabolic, and Volterra-type stochastic equations with Lévy noise, without boundedness restrictions on test functions.

Findings

Weak convergence rate is twice the strong rate for certain stochastic PDEs.

Framework applies to both hyperbolic and parabolic equations, including Volterra integro-differential equations.

Extended results to general square-integrable Lévy processes without bounded test function assumptions.

Abstract

We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive L\'evy noise. We first derive a representation formula for the error which we then apply to study space-time discretizations of the stochastic heat equation, a Volterra-type integro-differential equation, and the wave equation as examples. For twice continuously differentiable test functions with bounded second derivative (with an additional condition on the second derivative for the wave equation) the weak rate of convergence is found to be twice the strong rate. The results extend earlier work by two of the authors as we consider general square-integrable infinite-dimensional L\'evy processes and do not require boundedness of the test functions and their first derivative. Furthermore, the present framework is applicable to both hyperbolic and parabolic equations, and even to stochastic Volterra integro-differential equations.