Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs
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In this paper we prove convergence rates for time discretisation schemes for semi-linear 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 non-parabolic problems. The main results are optimal bounds for the uniform strong error E_k^∞ := (𝔼sup_j∈{0, …, N_k}U(t_j) - U^j^p)^1/p, where p ∈ [2,∞), U is the mild solution, U^j is obtained from a time discretisation scheme, k is the step size, and N_k = T/k. The usual schemes such as splitting/exponential Euler, implicit Euler, and Crank-Nicolson, etc. are included as special cases. Under conditions on the nonlinearity and the noise we show - E_k^∞≲ k log(T/k) (linear equation, additive noise, general S); - E_k^∞≲√(k)log(T/k) (nonlinear equation, multiplicative noise, contractive S); - E_k^∞≲ k log(T/k) (nonlinear wave equation, multiplicative noise). The logarithmic factor can be removed if the splitting scheme 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 error E_k:=(sup_j∈{0,…,N_k}𝔼U(t_j) - U^j^p)^1/p. 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.
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