# Robustness of nonuniform mean-square exponential dichotomies

## TL;DR

This paper proves that nonuniform mean-square exponential dichotomies in linear stochastic differential equations are stable under small perturbations, providing insights into their structure and robustness across different time domains.

## Contribution

It establishes the robustness of nonuniform mean-square exponential dichotomies for linear SDEs under small linear perturbations, including the analysis of related projections and contractions.

## Key findings

- NMS-ED persists under small perturbations
- Projections of solutions are related across time domains
- Results extend to NMS-EC

## Abstract

For linear stochastic differential equations (SDEs) with bounded coefficients, we establish the robustness of nonuniform mean-square exponential dichotomy (NMS-ED) on $[t_{0},+\oo)$, $(-\oo,t_{0}]$ and the whole $\R$ separately, in the sense that such an NMS-ED persists under a sufficiently small linear perturbation. The result for the nonuniform mean-square exponential contraction (NMS-EC) is also discussed. Moreover, in the process of proving the existence of NMS-ED, we use the observation that the projections of the "exponential growing solutions" and the "exponential decaying solutions" on $[t_{0},+\oo)$, $(-\oo,t_{0}]$ and $\R$ are different but related. Thus, the relations of three types of projections on $[t_{0},+\oo)$, $(-\oo,t_{0}]$ and $\R$ are discussed.