Pathwise uniform convergence of time discretization schemes for SPDEs

In this paper we prove convergence rates for time discretization schemes for semilinear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator A is the generator of a strongly continuous semigroup S on a Hilbert space X, and the focus is on nonparabolic problems. The main results are optimal bounds for the uniform strong erroralign* &Eₖ∞ := (E _j∈ \0, …, Nₖ\ \|U(tⱼ) - Uʲ\|ᵖ)1/p,align*where p ∈ [2,∞ ), U is the mild solution, Uʲ is obtained from a time discretization scheme, k is the step size and Nₖ = T/k. The usual schemes such as the exponential Euler (EE), the implicit Euler (IE), the Crank–Nicolson (CN) method, etc. are included as special cases. Under conditions on the nonlinearity and the noise, we show Eₖ∞ k √log (T/k) (linear equation, additive noise, general S)Eₖ∞ √k √log (T/k) (nonlinear equation, multiplicative noise, contractive S)Eₖ∞ k √log (T/k) (nonlinear wave equation, multiplicative noise), for a large class of time discretization schemes. The logarithmic factor can be removed if the EE method is used with a (quasi)-contractive S. The obtained bounds coincide with the optimal bounds for SDEs. Most of the existing literature is concerned with bounds for the simpler pointwise strong erroralign* &Eₖ := (_j∈ \0,…,Nₖ \|U(tⱼ) - Uʲ\|ᵖ)1/p.align*Applications to Maxwell equations, Schrödinger equations and wave equations are included. For these equations, our results improve and reprove several existing results with a unified method and provide the first results known for the IE and the CN method.