Stochastic perturbations of stable dynamical systems: trajectory-wise approach

We study stochastic perturbations of a dynamical system with a locally stable fixed point. The perturbed system has the form of Ito stochastic differential equations. We assume that perturbations do not vanish at the equilibrium of the deterministic system. Using the trajectory-wise approach to the analysis of stochastic differential equations, we find restrictions for perturbations under which the stability of the equilibrium is preserved with probability 1.