Options
Temporal approximation of stochastic evolution equations with irregular nonlinearities
Citation Link: https://doi.org/10.15480/882.13364
Publikationstyp
Journal Article
Date Issued
2024-05-09
Sprache
English
Author(s)
TORE-DOI
Journal
Volume
24
Issue
2
Article Number
43
Citation
Journal of Evolution Equations 24 (2): 43 (2024-05-09)
Publisher DOI
Scopus ID
Publisher
Springer
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 (Formula presented.) where p∈[2,∞), U is the mild solution, Uj is obtained from a time discretisation scheme, k is the step size, and Nk=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 (Formula presented.) 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.
Subjects
Low regularity
Pathwise uniform convergence
SPDEs
Stochastic convolutions
Time discretisation schemes
DDC Class
510: Mathematics
Publication version
publishedVersion
Loading...
Name
s00028-024-00975-6.pdf
Type
Main Article
Size
538.2 KB
Format
Adobe PDF