The Existence of Evolution Systems of Measures of Non-autonomous Stochastic Differential Equations with Infinite Delays

This paper is concerned with the dynamical behavior of nonautonomous stochastic differential equations with infinite delays. A new generalized Halanay inequality is introduced to estimate the solutions. By using the generalized Halanay inequality, we obtain that the mean-square of solution maps of such equations are exponentially attracted by a bounded set and are exponentially convergent from different initial data. Furthermore, the existence of the evolution system of measures for the stochastic equations and the stability in distribution of the evolution system of measures are also showed.