On the discretization in time of parabolic stochastic partial differential equations

We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion of order in probability and generalize in that context the results of the globally Lipschitz case. Mathematics Subject Classification. 60H15, 60F25, 60F99, 65C20, 60H35. Received: July 24, 2000. Revised: January 8, 2001; September 17, 2001.