  This paper studies a small-gain theorem for nonlinear stochastic equations
driven by additive white noise in both trajectories and stationary
distribution. Motivated by the most recent work of Marcondes de Freitas and
Sontag \cite{FS3}, we firstly define the {\it "input-to-state characteristic
operator"} $\mathcal{K}(u)$ of the system in a suitably chosen input space via
backward It\^o integral, and then for a given output function $h$, define the
$"gain\ operator"$ as the composition of output function $h$ and the
input-to-state characteristic operator $\mathcal{K}(u)$ on the input space.
Suppose that the output function is either order-preserving or
anti-order-preserving in the usual vector order and the global Lipschitz
constant of the output function is less than the absolute of the negative
principal eigenvalue of linear matrix. Then we prove the so-called {\it
"small-gain theorem"}: the gain operator has a unique fixed point, the image
for input-to-state characteristic operator at the fixed point is a globally
attracting stochastic equilibrium for the random dynamical system generated by
the stochastic system. Under the same assumption for the relation between the
Lipschitz constant of the output function and maximal real part of stable
linear matrix, we prove that the stochastic system has a unique stationary
distribution, which is regarded as a stationary distribution version of
small-gain theorem. These results can be applied to stochastic cooperative,
competitive and predator-prey systems, or even others.
