Temporal approximation of stochastic evolution equations with irregular nonlinearities

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 errorEk∞:=(Esupj∈{0,…,Nk}‖U(tj)-Uj‖Xp)1/p→0(k→0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_{k}^{\\infty } {:}{=}\\Big (\\mathbb {E}\\sup _{j\\in \\{0, \\ldots , N_k\\}} \\Vert U(t_j) - U^j\\Vert _X^p\\Big )^{1/p} \\rightarrow 0\\quad (k \\rightarrow 0), \\end{aligned}$$\\end{document}where p∈[2,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in [2,\\infty )$$\\end{document}, U is the mild solution, Uj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U^j$$\\end{document} is obtained from a time discretisation scheme, k is the step size, and Nk=T/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_k = T/k$$\\end{document} for final time T>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T>0$$\\end{document}. 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 errorEk:=(supj∈{0,…,Nk}E‖U(tj)-Uj‖Xp)1/p,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_k {:}{=}\\bigg (\\sup _{j\\in \\{0,\\ldots ,N_k\\}}\\mathbb {E}\\Vert U(t_j) - U^{j}\\Vert _X^p\\bigg )^{1/p}, \\end{aligned}$$\\end{document}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.