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Weak order for the discretization of the stochastic heat equation

Authors: Arnaud Debussche and Jacques Printems
Journal: Math. Comp. 78 (2009), 845-863
MSC (2000): Primary 60H15, 60H35, 65C30, 65M60
Published electronically: October 7, 2008
MathSciNet review: 2476562
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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

$\displaystyle \mathrm{d} X_t+AX_t \, \mathrm{d} t = Q^{1/2} \mathrm{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 a $ \theta$-method 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

$\displaystyle \vert\mathbb{E} \, \varphi(X^N_h) - \mathbb{E} \, \varphi(X_T) \vert = O(h^{2\gamma} + \Delta t^\gamma) $

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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Additional Information

Arnaud Debussche
Affiliation: IRMAR et ENS de Cachan, antenne de Bretagne, Campus de Ker Lann, avenue Robert Schumann, 35170 Bruz, France

Jacques Printems
Affiliation: Laboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, Université de Paris XII, 61, avenue du Général de Gaulle, 94010 Créteil, France

Keywords: Weak order, stochastic heat equation, finite element, Euler scheme
Received by editor(s): October 30, 2007
Received by editor(s) in revised form: May 7, 2008
Published electronically: October 7, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.