Weak convergence of finite element approximations of linear stochastic evolution equations with additive Lévy noise
For researchers in numerical analysis of stochastic PDEs, this work extends prior results to more general Lévy processes and equations, but is incremental in nature.
The paper develops an abstract framework for analyzing weak convergence of numerical approximations for linear stochastic PDEs with additive Lévy noise, proving that the weak rate is twice the strong rate for certain test functions. Examples include the stochastic heat equation, a Volterra-type equation, and the wave equation.
We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive Lévy 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 Lévy 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.