Temporal approximation of stochastic evolution equations with irregular nonlinearities
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In this paper we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error E_k^∞ := (𝔼sup_j∈{0, …, N_k}U(t_j) - U^j_X^p)^1/p→ 0 (k → 0), 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 for final time T>0. This generalises previous results to a larger class of admissible nonlinearities and noise as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error E_k := (sup_j∈{0,…,N_k}𝔼U(t_j) - U^j_X^p)^1/p, which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.
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