Overall Objectives
Scientific Foundations
Application Domains
New Results
Other Grants and Activities

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

Im55 $\mtable{...}$

driven by a Gaussian space time noise whose covariance operator Q is given. We assume that Im56 $A^{-\#945 }$ is a finite trace operator for some $ \alpha$>0 and that Q is bounded from H into Im57 ${D(A^\#946 )}$ for some $ \beta$$ \ge$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 $ \upper_delta$t = T/N ). We define a discrete solution XhN and for suitable functions Im58 $\#981 $ defined on H , we show that

Im59 $\mtable{...}$

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.


Logo Inria