Generalized Ornstein-Uhlenbeck process for affine stochastic functional differential equations and its applications

This paper studies the existence and global stability of generalized Ornstein-Uhlenbeck process for affine stochastic functional differential equations. Various very basic and important properties are established. In the applications, we present a standard and rigorous procedure for guaranteeing the existence and uniqueness of random equilibria for nonlinear stochastic functional differential equations, which attracts all pull-back trajectories in different types of convergence. Some examples are given to illustrate our main results. The results presented in this paper improve and simplify the conclusions of Jiang and Lv [{\it SIAM J. Control Optim.}, 54 (2016), pp. 2383-2402] and [{\it J. Differential Equations}, 367 (2023), pp. 890-921].