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

Publikation: Beitrag in FachzeitschriftForschungsartikelBeigetragenBegutachtung



We study the approximation of the distribution of XT, where (Xt)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, dXt+AXtdt = Q1/2dZt, X0=x0 ∈ H, t ∈ [0,T]. Here (Zt)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 AβQ is bounded for some β ∈ (α - 1, α]. A discretized solution (Xhn)n∈{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 ∈ C2b(H;ℝ) we show an integral representation for the error {pipe}Eφ(XNh)-Eφ(XT){pipe} and prove that, where γ < 1 - 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.


Seiten (von - bis)345-379
FachzeitschriftPotential Analysis
PublikationsstatusVeröffentlicht - Feb. 2013


ASJC Scopus Sachgebiete


  • Euler scheme, Finite element, Impulsive cylindrical process, Infinite dimensional Lévy process, Stochastic heat equation, Weak order