On the weak order of the Euler Schemes for stochastic Partial Differential Equations

In this talk, we present recent results on the order of convergence of the Euler scheme for a Stochastic Partial Differential Equation. The strong order of convergence has been studied by many authors. However, very few results are available for the weak order of convergence.

It is well known that the Euler scheme is of strong order $1/2$ and weak order $1$ in the case of a stochastic differential equation. Two
methods are available to prove this result. The first one uses the Kolmogorov equation associated to the stochastic equation and was first used by D. Talay. A second one has been recently discovered by A. Kohatsu-Higa and is based on Malliavin calculus.

In this talk, we generalize such results to the infinite dimensional case. We show how to adapt Talay s method. The main difficulty is due
to the presence of unbounded operators in the Kolmogorov equation. A tricky change of unknown allows to treat the case of a linear equation. It also works for an equation whose linear part defines a group, the nonlinear Schr odinger equation for instance. The case of a semilinear equation of parabolic type is more difficult and we use Malliavin calculus, but not in the same way as in Kohatsu-Higa s method. We prove for instance that, in the case of a nonlinear heat
equation in dimension one with a space time white noise, the Euler scheme has weak order $1/2$, it is well known that the strong order is $1/4$.