Weak order for the discretization of the stochastic heat equation

Source abstract

In this paper we study the approximation of the distribution of X t X_t Hilbert–valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$d X t + A X t d t = Q 1 / 2 d W ( t ) , X 0 = x ∈ H , t ∈ [ 0 , T ] , \mathrm {d} X_t+AX_t \, \mathrm {d} t = Q^{1/2} \mathrm {d} W(t), \quad X_0=x \in H, \quad t\in [0,T],$$ driven by a Gaussian space time noise whose covariance operator Q Q is given. We assume that A − α A^{-\alpha } is a finite trace operator for some α > 0 \alpha >0 and that Q Q is bounded from H H into D ( A β ) D(A^\beta ) for some β ≥ 0 \beta \geq 0 . It is not required to be nuclear or to commute with A A . The discretization is achieved thanks to finite element methods in space (parameter h > 0 h>0 ) and a θ \theta -method in time (parameter Δ t = T / N \Delta t=T/N ). We define a discrete solution X h n X^n_h and for suitable functions φ \varphi defined on H H , we show that $$| E φ ( X h N ) − E φ ( X T ) | = O ( h 2 γ + Δ t γ ) |\mathbb {E} \, \varphi (X^N_h) - \mathbb {E} \, \varphi (X_T) | = O(h^{2\gamma } + \Delta t^\gamma )$$ where γ > 1 − α + β \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.