Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise

We study the approximation of the distribution of X _T , where (X _t ) _t?∈?[0,?T] is a Hilbert space valued stochastic process that solves a linear parabolic stochastic partial differential equation driven by an impulsive space time noise, $$ dX_t+AX_t,dt= Q^{1/2},dZ_t,quad X_0=x_0in H,quad tin [0,T]. $$ Here (Z _t ) _t?∈?[0,?T] is an impulsive cylindrical process and the operator Q describes the spatial covariance structure of the noise; we assume that A ^??α has finite trace for some α?>?0 and that A ^β Q is bounded for some β?∈?(α???1, α]. A discretized solution $(X_h^n)_{nin{0,1,ldots,N}}$ is defined via the finite element method in space (parameter h?>?0) and a θ-method in time (parameter Δt?=?T/N). For $varphi in C^2_b(H;{mathbb R})$ we show an integral representation for the error $|{mathbb E}varphi(X^N_h)-{mathbb E}varphi(X_T)|$ and prove that $$ left|{mathbb E}varphileft(X^N_hright)-{mathbb E}varphi(X_T)right|=Oleft(h^{2gamma}+left(Delta tright)^{gamma}right) $$ where γ?α?+?β. This is the same order of convergence as in the case of a Gaussian space time noise, which has been obtained in a paper by Debussche and Printems (Math Comput 78:845–863, 2009). Our result also holds for a combination of impulsive and Gaussian space time noise.