Singular Perturbations of Degenerate Diffusions

Abstract

Ventsel and Freidlin studied small random perturbations $dx^\varepsilon(t) = b(x^\varepsilon(t)) dt + \sqrt{\varepsilon} \sigma (x^\varepsilon(t)) d\beta(t)$ of the dynamical system $dx(t) = b(x(t)) dt$. They proved a large deviations theorem for the family $\{x^\varepsilon(t)\}$, and used the large deviations estimates to prove (among other things) a result about exit from a bounded domain containing an equilibrium of the unperturbed system. Fleming rederived this exit theorem using techniques from stochastic control theory. In this paper we study analogous questions for singular perturbations of degenerate diffusions. We consider the family of processes $dx^\varepsilon(t) = b(x^\varepsilon(t)) dt + \tau(x^\varepsilon(t))\circ dz(t) + \sqrt{\varepsilon} \sigma(x^\varepsilon(t)) d\beta(t)$ for $\varepsilon \geq 0$, where the process $x^0(t)$ is degenerate. We show that a large deviations principle need not hold, but we derive bounds sufficient for obtaining some estimates on probabilities. We also adapt Fleming's approach to prove an exit theorem.