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 \[ \mathrm {d} X_t+AX_t \mathrm {d} t = Q^{1/2} \mathrm {d} W(t), \quad X_0=x \in H, \quad t\in [0,T], \] 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 \[ |\mathbb {E} \varphi (X^N_h) - \mathbb {E} \varphi (X_T) | = O(h^{2\gamma } + \Delta t^\gamma ) \] 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.