Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations

TL;DR

This paper investigates the existence, uniqueness, and large deviation properties of solutions for stochastic Volterra equations with singular kernels in Banach spaces, and applies these results to various classes of stochastic partial differential equations, including Navier-Stokes.

Contribution

It introduces new methods for analyzing stochastic Volterra equations with singular kernels and extends these techniques to a broad range of SPDEs, including high order and geometric cases.

Findings

Existence and uniqueness of solutions for stochastic Volterra equations with singular kernels.

Application of these results to semilinear SPDEs driven by Brownian and fractional Brownian motions.

Analysis of high order SPDEs, SPDEs on manifolds, and stochastic Navier-Stokes equations.

Abstract

In this paper, we first study the existence-uniqueness and large deviation estimate of solutions for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then, we apply them to a large class of semilinear stochastic partial differential equations (SPDE) driven by Brownian motions as well as by fractional Brownian motions, and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Lastly, high order SPDEs in a bounded domain of Euclidean space, second order SPDEs on complete Riemannian manifolds, as well as stochastic Navier-Stokes equations are investigated.