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

TL;DR

This paper introduces a generalized Ornstein-Uhlenbeck process for affine stochastic functional differential equations, establishing existence, stability, and convergence properties, with applications to random equilibria and improvements over prior results.

Contribution

It develops a unified framework for the generalized Ornstein-Uhlenbeck process in affine stochastic functional differential equations, simplifying and extending previous theoretical results.

Findings

Established existence and stability of the generalized Ornstein-Uhlenbeck process.

Provided a rigorous method for guaranteeing unique random equilibria.

Illustrated main results with concrete examples.

Abstract

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].