Stochastic Volterra equations driven by Gaussian noise
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 $\alpha \in (0,2)$. This presentation is based on a joint work with Luigi Amedeo Bianchi (Trento) and Martin Friesen (Dublin City University).
Area: CS7 - Advances in SPDEs (Giuseppina Guatteri and Federica Masiero)
Keywords: Stochastic Volterra equations, completely monotone kernels, limit distributions
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