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

A small-gain theorem for nonlinear stochastic systems with inputs and outputs I: Additive white noise