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July, 1987 Singular Perturbations of Degenerate Diffusions
Carol Bezuidenhout
Ann. Probab. 15(3): 1014-1043 (July, 1987). DOI: 10.1214/aop/1176992078


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.


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Carol Bezuidenhout. "Singular Perturbations of Degenerate Diffusions." Ann. Probab. 15 (3) 1014 - 1043, July, 1987.


Published: July, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0646.60031
MathSciNet: MR893911
Digital Object Identifier: 10.1214/aop/1176992078

Primary: 60F10
Secondary: 34F05 , 35B25 , 60H10 , 60H25 , 60J60

Keywords: Degenerate diffusions , Exit problem , large deviations , singular perturbations , stochastic control theory , Ventsel-Freidlin theory

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 3 • July, 1987
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