Pathwise convergence of an efficient scheme for SPDEs with non-globally Lipschitz nonlinearity

摘要

This paper aims to extend the scheme proposed in Jentzen et al. (2011) for stochastic partial differential equations (SPDEs) with global Lipschitz coefficients to non-global Lipschitz coefficients. We investigate the pathwise convergence of the scheme for a class of semilinear parabolic SPDEs with non-globally Lipschitz nonlinearity. We show first that the scheme is convergent uniformly in time and space under Lipschitz assumption on the nonlinearity of the SPDE, then we obtain the convergence in the case of non-globally Lipschitz nonlinearity via a localization technique. Compared to the scheme introduced in Jentzen (2009) for SPDEs under non-global Lipschitz coefficients, the scheme considered in this paper is simpler in the sense that the former uses two linear functionals of the noise while the latter uses one.