The Samoilenko reduction principle for differential equations with random perturbations (Q5951219)

Let \(x(t,t_0,x_0)\) be a solution to the randomly perturbed differential equation \[ dx/dt=F(x) +\sigma(t,x)\xi(f),\quad t\geq 0,\tag{*} \] with \(x\in \mathbb{R}^n\), \(\xi(t)\) a random process a.s. absolutely integrable on every finite interval, and \(F,\sigma \in\text{Lip}\). The set \(S_t\) is said to be positively invariant if \[ P\{x(t,t_0,x_0)\in S_t,\quad t\geq 0\}=1. \] The author considers the stability of a positively invariant set on which system (*) is reduced to the corresponding deterministic one.