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Weak order for the discretization of the stochastic heat equation.

Abstract : In this paper we study the approximation of the distribution of $X_t$ Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$ dX_t+AX_t \, dt = Q^{1/2} d W_t, \quad X_0=x \in H, \quad t\in[0,T], $$ driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-\alpha}$ is a finite trace operator for some $\alpha>0$ and that $Q$ is bounded from $H$ into $D(A^\beta)$ for some $\beta\geq 0$. It is not required to be nuclear or to commute with $A$. The discretization is achieved thanks to finite element methods in space (parameter $h>0$) and implicit Euler schemes in time (parameter $\Delta t=T/N$). We define a discrete solution $X^n_h$ and for suitable functions $\varphi$ defined on $H$, we show that $$ |\E \, \varphi(X^N_h) - \E \, \varphi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) $$ \noindent where $\gamma<1- \alpha + \beta$. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations.
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Contributor : Marie-Annick Guillemer <>
Submitted on : Monday, October 29, 2007 - 4:02:35 PM
Last modification on : Friday, July 10, 2020 - 4:17:23 PM
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  • HAL Id : hal-00183249, version 1
  • ARXIV : 0710.5450


Arnaud Debussche, Jacques Printems. Weak order for the discretization of the stochastic heat equation.. Mathematics of Computation, American Mathematical Society, 2009, 78 (266), pp.845-863. ⟨hal-00183249⟩



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