# Nonuniform Mean-square Exponential Dichotomies and Mean-square Exponential Stability

## TL;DR

This paper establishes conditions for nonuniform mean-square exponential dichotomy in linear stochastic differential equations and explores mean-square exponential stability of their nonlinear perturbations, introducing the second-moment regularity coefficient.

## Contribution

It introduces the concept of second-moment regularity coefficient and provides new criteria for stability and dichotomy in stochastic differential equations, extending deterministic results.

## Key findings

- Established existence conditions for NMS-ED in SDEs.
- Introduced the second-moment regularity coefficient for stability analysis.
- Derived bounds for the coefficient based on the drift term.

## Abstract

In this paper, the existence conditions of nonuniform mean-square exponential dichotomy (NMS-ED) for a linear stochastic differential equation (SDE) are established. The difference of the conditions for the existence of a nonuniform dichotomy between an SDE and an ordinary differential equation (ODE) is that the first one needs an additional assumption, nonuniform Lyapunov matrix, to guarantee that the linear SDE can be transformed into a decoupled one, while the second does not. Therefore, the first main novelty of our work is that we establish some preliminary results to tackle the stochasticity. This paper is also concerned with the mean-square exponential stability of nonlinear perturbation of a linear SDE under the condition of nonuniform mean-square exponential contraction (NMS-EC). For this purpose, the concept of second-moment regularity coefficient is introduced. This concept is essential in determining the stability of the perturbed equation, and hence we deduce the lower and upper bounds of this coefficient. Our results imply that the lower and upper bounds of the second-moment regularity coefficient can be expressed solely by the drift term of the linear SDE.