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Register a new userRobustness of nonuniform mean-square exponential dichotomies
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Hailong Zhu
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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.
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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.
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