Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schr\"odinger and linearized stochastic Korteweg-de Vries equations

TL;DR

This paper derives sharp weak convergence rates for noise discretizations in a broad class of stochastic evolution equations, including nonlinear wave, Schrödinger, and KdV equations, improving upon previous suboptimal results.

Contribution

It provides the first essentially sharp weak convergence rates for several important stochastic PDEs with non-regularizing semigroups and various noise types.

Findings

Established sharp weak convergence rates for stochastic wave equations

Extended results to HJMM, Schrödinger, and KdV equations

Improved upon previous suboptimal convergence rate estimates

Abstract

We establish weak convergence rates for noise discretizations of a wide class of stochastic evolution equations with non-regularizing semigroups and additive or multiplicative noise. This class covers the nonlinear stochastic wave, HJMM, stochastic Schr\"odinger and linearized stochastic Korteweg-de Vries equation. For several important equations, including the stochastic wave equation, previous methods give only suboptimal rates, whereas our rates are essentially sharp.