Stochastic evolution equations with Volterra noise

Volterra processes are continuous stochastic processes whose covariance function can be written in the form R(s,t)=∫0s∧tK(s,r)K(t,r)dr, where K is a suitable square integrable kernel. Examples of such processes are the fractional Brownian motion, multifractional Brownian motion or (in the non-Gaussian case) Rosenblatt process. In the first part, stochastic integral with respect to Volterra processes and cylindrical Volterra process in Hilbert spaces are defined and some of their properties are studied. In the second part, these results are applied to linear stochastic equations in Hilbert spaces driven by cylindrical Volterra processes. Measurability, mean-square continuity and paths continuity of their solutions are proved under various sets of conditions. The general results are illustrated by examples of parabolic and hyperbolic SPDEs.