Weak order for the discretization of the stochastic heat equation driven by impulsive noise
Provides the first weak error analysis for SPDEs with impulsive noise, extending known results from Gaussian to impulsive settings.
The paper studies the weak error of discretizing a stochastic heat equation driven by impulsive noise, proving an error bound of O(h^{2γ} + (Δt)^γ) with γ < 1 - α + β for the finite element and θ-method discretization.
Considering a linear parabolic stochastic partial differential equation driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H, t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an impulsive cylindrical process and Q describes the spatial covariance structure of the noise; Tr(A^{-α})<\infty for some α>0 and A^βQ is bounded for some β\in(α-1,α]. A discretization (X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space (parameter h>0) and a θ-method in time (parameter Δt=T/N). For ϕ\in C^2_b(H;R) we show an integral representation for the error |Eϕ(X^N_h)-Eϕ(X_T)| and prove that |Eϕ(X^N_h)-Eϕ(X_T)|=O(h^{2γ}+(Δt)^γ) where γ<1-α+β.