Weak order for the discretization of the stochastic heat equation

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 \[ d X t + A X t d t = Q 1 / 2 d W ( t ) , X 0 = x ∈ H , t ∈ [ 0 , T ] , d X_t+AX_t \, d t = Q1/2 d W(t), X_0=x ∈ H, t∈ [0,T], \] driven by a Gaussian space time noise whose covariance operator Q Q is given. We assume that A − α A-α is a finite trace operator for some α > 0 α >0 and that Q Q is bounded from H H into D ( A β ) D(A^β ) for some β ≥ 0 β ≥ 0 . It is not require