Robustness of nonuniform mean-square exponential dichotomies

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 $\mathbb{R}$ 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 $\mathbb{R}$ are different but related. Thus, the relations of three types of projections on $[t_{0},+\oo)$, $(-\oo,t_{0}]$ and $\mathbb{R}$ are discussed.