Weak error estimates for trajectories of SPDEs for Spectral Galerkin discretization

Abstract

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection, and we study the approximation of the probability distribution of the trajectories: test functions are regular, but depend on the values of the process on the interval $[0,T]$. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Itô map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Itô map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

References in corpus (5)

• Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE
• Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
• Weak error analysis for semilinear stochastic Volterra equations with additive noise
• Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise

Cited by in corpus (4)

• A full-discrete exponential Euler approximation of invariant measure for parabolic stochastic partial differential equations
• Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations
• Stochastic Discontinuous Galerkin Methods (SDGM) Based on Fluctuation-Dissipation Balance
• Weak error analysis via functional Itô calculus