Dynamical systems under the action of fast random perturbations (Q809474)

The author considers dynamical systems of the type \[ dx_{\epsilon}(t)/dt=a(x_{\epsilon}(t),y_{\epsilon}(t)),\quad \epsilon >0. \] The pair \((x_{\epsilon}(t),y_{\epsilon}(t))\) is a Markov process with the infinitesimal operator \[ A_{\epsilon}f(x,y)=(a(x,y),f_ x(x,y))+(1/\epsilon)\int [f(x,y')- f(x,y)]\Pi (x,y,dy'), \] where \(\Pi\) (x,y,\(\cdot)\) is a finite measure. Sufficient conditions are given under which the processes \(x_{\epsilon}(t/\epsilon)\) converge weakly to a certain diffusion process as \(\epsilon\to 0\).