On Parabolic Volterra Equations Disturbed by Fractional Brownian Motions

Abstract

Aim of this article is to study the parabolic Volterra equation on a separable Hilbert space. Throughout this work the operator −A is assumed to be a differential operator like the Laplacian, the elasticity operator, or the Stokes operator. The random disturbance Q 1/2 is modeled to be a system independent vector valued fractional Brownian motion with Hurst parameter ∈ (0, 1). We derive optimal conditions for the existence of a unique mild solution and the Hölderianity of its trajectories. For this purpose we do the analysis on stochastic integrals of the form where the integrand R is a deterministic, operator valued function.