Mixing and weak convergence of numerical approximations of SPDEs

Abstract: We consider a general framework for obtaining uniform-in-time rates of convergence for numerical approximations of SPDEs in suitable Wasserstein distances. The framework is based on two general results under an appropriate set of assumptions: a Wasserstein contraction result for a given Markov semigroup; and a uniform-in-time weak convergence result for a parametrized family of Markov semigroups. We provide an application to a suitable space-time discretization of the 2D stochastic Navier-Stokes equations in vorticity formulation. Specifically, we obtain that the Markov semigroup induced by this discretization satisfies a Wasserstein contraction result which is independent of any discretization parameters. This allows us to obtain a corresponding weak convergence result towards the Markov semigroup induced by the 2D SNSE. The proof required technical improvements from the related literature regarding finite-time error estimates. Finally, our approach does not rely on standard gradient estimates for the underlying Markov semigroup, and thus provides a flexible formulation for further applications.

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