  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.
