Existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions

TL;DR

This paper develops a comprehensive framework for establishing existence and uniqueness of solutions to stochastic functional differential equations with delays in infinite-dimensional spaces, applicable to various SPDEs.

Contribution

It introduces a general approach for martingale solutions and proves strong solution uniqueness under local monotonicity conditions for delayed SPDEs.

Findings

Framework applicable to a wide class of SPDEs with delays

Existence of martingale solutions for these equations

Uniqueness of strong solutions under specific conditions

Abstract

In this paper, we present a general framework for solving stochastic functional differential equations in infinite dimensions in the sense of martingale solutions, which can be applied to a large class of SPDE with finite delays, e.g. $d$-dimensional stochastic fractional Navier-Stokes equations with delays, $d$-dimensional stochastic reaction-diffusion equations with delays, $d$-dimensional stochastic porous media equations with delays. Moreover, under local monotonicity conditions for the nonlinear term we obtain the existence and uniqueness of strong solutions to SPDE with delays.