Robustness of nonuniform mean-square exponential dichotomies

Abstract

For linear stochastic differential equations with bounded coefficients, we establish the robustness of nonuniform mean-square exponential dichotomy (NMS-ED) on $[t_{0},\,+\infty )$ , $(-\infty,\,t_{0}]$ and the whole ${\Bbb R}$ 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 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 $[t_{0},\,+\infty )$ , $(-\infty,\,t_{0}]$ and ${\Bbb R}$ are different but related. Thus, the relations of three types of projections on $[t_{0},\,+\infty )$ , $(-\infty,\,t_{0}]$ and ${\Bbb R}$ are discussed.