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- W4384652307 abstract "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 $$E_{k}^{infty} := Big(mathbb{E} sup_{jin {0, ldots, N_k}} |U(t_j) - U^j|_X^pBig)^{1/p} to 0quad (k to 0),$$ where $p in [2,infty)$, $U$ 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 $$E_k := bigg(sup_{jin {0,ldots,N_k}}mathbb{E} |U(t_j) - U^{j}|_X^pbigg)^{1/p},$$ which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrodinger equation, for which previous convergence results were not applicable." @default.
- W4384652307 created "2023-07-19" @default.
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- W4384652307 date "2023-07-14" @default.
- W4384652307 modified "2023-10-18" @default.
- W4384652307 title "Temporal approximation of stochastic evolution equations with irregular nonlinearities" @default.
- W4384652307 doi "https://doi.org/10.48550/arxiv.2307.07596" @default.
- W4384652307 hasPublicationYear "2023" @default.
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