Abstract: 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 is the generator of a strongly continuous semigroup on a Hilbert space , and the focus is on non-parabolic problems. The main results are optimal bounds for the uniform strong error mathrm{E}_{k}^{infty} := Big(mathbb{E} sup_{jin {0, ldots, N_k}} |U(t_j) - U^j|^pBig)^{1/p}, where , is the mild solution, is obtained from a time discretisation scheme, is the step size, and . 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 - (linear equation, additive noise, general ); - (nonlinear equation, multiplicative noise, contractive ); - (nonlinear wave equation, multiplicative noise). The logarithmic factor can be removed if the splitting scheme is used with a (quasi)-contractive . 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 mathrm{E}_k:=�igg(sup_{jin {0,ldots,N_k}}mathbb{E} |U(t_j) - U^{j}|^p�igg)^{1/p}. Applications to Maxwell equations, Schr"odinger equations, and wave equations are included. For these equations our results improve and reprove several existing results with a unified method.
