Global stability of stationary solutions for a class of semilinear stochastic functional differential equations with additive white noise

Abstract

This paper gives a criterion for the existence of a stationary solution for a class of semilinear stochastic functional differential equations with additive white noise and its global stability. Under the condition that the global Lipschitz constant of nonlinear term f is less than the absolute value of the top Lyapunov exponent for the linear flow Φ with f being monotone or anti-monotone, and the time delay is not very big, we show that the infinite-dimensional stochastic flow possesses a unique globally attracting random equilibrium in the state space of continuous functions, which produces the globally stable stationary solution. Compared to the result of Jiang and Lv (2016) [24], we remove the assumption of boundedness for f.