On the Almost Sure Stability of Dynamical Systems with Respect to White Noise

We consider a system of ordinary differential equations possessing an asymptotically stable equilibrium position and examine the stability of this equilibrium under permanent stochastic white-noise-type perturbations. The perturbed system is considered in the form of Stratonovich stochastic differential equations. We also describe restrictions on the parameters of perturbations that guarantee the trajectory-wise stability of the equilibrium with probability $1$.