Limits of stochastic Volterra equations driven by Gaussian noise

Abstract

We study stochastic Volterra equations in Hilbert spaces driven by cylindrical Gaussian noise. We derive a mild formulation for the stochastic Volterra equation, prove the equivalence of mild and strong solutions, the existence and uniqueness of mild solutions, and study space-time regularity. Furthermore, we establish the stability of mild solutions in $L^q({\mathbb{R}}_{+})$, prove the existence of limit distributions in the Wasserstein $p$-distance with $p\in [1,\infty )$, and characterise when these limit distributions are independent of the initial state of the process despite the presence of memory. While our techniques allow for a general class of Volterra kernels, they are particularly suited for completely monotone kernels and fractional Riemann-Liouville kernels in the full range $\alpha \in (0,2)$.