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We study stochastic Volterra equations in Hilbert spaces driven bycylindrical Gaussian noise. We derive a mild formulation for the stochasticVolterra equation, prove the equivalence of mild and strong solutions, theexistence and uniqueness of mild solutions, and study space-time regularity.Furthermore, we establish the stability of mild solutions in $L^q(\R_+)$, provethe existence of limit distributions in the Wasserstein $p$-distance with $p\in [1,\infty)$, and characterise when these limit distributions areindependent of the initial state of the process despite the presence of memory.While our techniques allow for a general class of Volterra kernels, they areparticularly suited for completely monotone kernels and fractionalRiemann-Liouville kernels in the full range $\alpha \in (0,2)$.

Limits of stochastic Volterra equations driven by Gaussian noise