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 Xt 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 XhN 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.
