Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients
Daoyi Xu Weisong Zhou
Discrete & Continuous Dynamical Systems - A 2017, 37(4): 2161-2180 doi: 10.3934/dcds.2017093

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.

keywords: Stochastic PDEs pathwise solutions existence uniqueness exponential estimate
Existence theorems for periodic Markov process and stochastic functional differential equations
Daoyi Xu Yumei Huang Zhiguo Yang
Discrete & Continuous Dynamical Systems - A 2009, 24(3): 1005-1023 doi: 10.3934/dcds.2009.24.1005
In this paper, an effective existence theorem for periodic Markov process is first established. Using the theorem, we consider a class of periodic $It\hat{o}$ stochastic functional differential equations, and some sufficient conditions for the existence of periodic solution of the equations are given. To overcome the difficulties created by the special features possessed by the periodic stochastic differential equations with delays, as one will see, several lemmas are introduced. These existence theorems are rather general and therefore have great power in applications. Especially, our results are natural generalization of some classical periodic theorems on the model without stochastic perturbation. An example is worked out to demonstrate the advantages of our results.
keywords: Attracting set. point dissipativity Stochastic differential equations Periodic solution Delays Markov process

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