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-$p$ Hölder continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space-time (Sobolev-Hölder) 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.

References in corpus (4)

• Optimal Regularity for Semilinear Stochastic Partial Differential Equations with Multiplicative Noise
• Convolution type stochastic Volterra equations
• Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term
• Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation

Cited by in corpus (6)

• Weak error analysis for semilinear stochastic Volterra equations with additive noise
• Mittag-Leffler Euler integrator for a stochastic fractional order equation with additive noise
• Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains
• Asymptotic properties of some space-time fractional stochastic equations
• Backward problem for time fractional reaction-diffusion equation with nonlinear source and discrete data
• Global solutions to the stochastic Volterra Equation driven by Lévy noise