Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise
Abstract
We consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under appropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean- H\"older continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space-time (Sobolev-H\"older) regularity of the solution together with a maximal type bound on the spatial Sobolev norm. As one of the main technical tools we establish a smoothing property of the derivative of the deterministic evolution operator family.
Keywords
Cite
@article{arxiv.1404.4131,
title = {Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise},
author = {Boris Baeumer and Matthias Geissert and Mihaly Kovacs},
journal= {arXiv preprint arXiv:1404.4131},
year = {2016}
}