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

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. dd-dimensional stochastic fractional Navier–Stokes equations with delays, dd-dimensional stochastic reaction–diffusion equations with delays, dd-dimensional stochastic porous media equations with delays. Moreover, under local monotonicity conditions for the nonlinear terms we obtain the existence and uniqueness of strong solutions to SPDE with delays.