## Section: New Results

### Weak order for the discretization of the stochastic heat equation

Participant : Arnaud Debussche.

In this joint work [22] with J. Printems (Université Paris XII),
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

driven by a Gaussian space time noise whose covariance operator Q is given. We assume that is a finite trace operator for some >0 and that Q is bounded from H into for some 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 t = T/N ). We define a discrete solution X_{h}^{N} and
for suitable functions defined on H , we show that

where <1- + . Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations.