Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients

Abstract

In this paper, we study the existence-uniqueness and exponential estimate of the pathwise mild solution of retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. Firstly, the existence-uniqueness of the maximal local pathwise mild solution are given by the generalized local Lipschitz conditions, which extend a classical Pazy theorem on PDEs. We assume neither that the noise is given in additive form or that it is a very simple multiplicative noise, nor that the drift coefficient is global Lipschitz continuous. Secondly, the existence-uniqueness of the global pathwise mild solution are given by establishing an integral comparison principle, which extends the classical Wintner theorem on ODEs. Thirdly, an exponential estimate for the pathwise mild solution is obtained by constructing a delay integral inequality. Finally, the results obtained are applied to a retarded stochastic infinite system and a stochastic partial functional differential equation. Combining some known results, we can obtain a random attractor, whose condition overcomes the disadvantage in existing results that the exponential converging rate is restricted by the maximal admissible value for the time delay.