Exit Problem and Stochastic Resonance for a Class of Random Perturbations

dc.contributor.advisorFreidlin, Mark I.en_US
dc.contributor.authorYang, zhihuien_US
dc.contributor.departmentMathematicsen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2005-08-03T14:21:31Z
dc.date.available2005-08-03T14:21:31Z
dc.date.issued2005-04-20en_US
dc.description.abstractThe asymptotic exit problems for diffusion processes with small parameter were considered in the classic work of Freidlin and Wentzell. In 2000, a mathematical theory of stochastic resonance for systems with random perturbations was established by Freidlin in the frame of the large deviation theory. This dissertation concerns exit problems and stochastic resonance for a class of random perturbations approximating white noise. The tools used in the proofs are the large deviation theory and the Markov property of the processes. The first problem considered is the exit problem and stochastic resonance for random perturbations of random walks. It turns out that a specific random walk can be chosen which approximates the large deviation asymptotics of the Wiener process in the best way. Analogous results concerning exit problems and stochastic resonance for this type of random perturbations were obtained under appropriate assumptions and compared with those of white noise type perturbation. The second problem I consider is the exit problems for random perturbations of a Gaussian process $\eta_{t}^{\mu,\varepsilon}$ which satisfies the equation $ \mu \dot{\eta}_{t}^{\mu,\varepsilon}=- \eta_{t}^{\mu,\varepsilon}+\sqrt{\varepsilon}\dot{W}_{t}, \,\eta_{0}^{\mu,\varepsilon}=y, \,0<\mu<<1,\,0<\varepsilon<<1 $. One can check that $\int_{0}^{t} \eta_{s}^{\mu,\varepsilon}ds$ converges to $\sqrt{\varepsilon}W_{t}$ uniformly on $[0,T]$ in probability as $\mu \downarrow 0$. Results concerning asymptotic exit problems for this type of random perturbation were obtained under appropriate assumptions. Since $\eta_{t}^{\mu,\varepsilon}$ is not a Markov process, this creates some difficulties for the proof. A new Markov process was constructed and the Markov property of the new process was used in the proof.en_US
dc.format.extent430818 bytes
dc.format.extent430818 bytes
dc.format.mimetypeapplication/pdf
dc.format.mimetypeapplication/pdf
dc.identifier.urihttp://hdl.handle.net/1903/2474
dc.language.isoen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pqcontrolledStatisticsen_US
dc.subject.pquncontrolledexit problemsen_US
dc.subject.pquncontrolledstochastic resonanceen_US
dc.subject.pquncontrolledlarge deviation theoryen_US
dc.subject.pquncontrolledrandom perturbationen_US
dc.subject.pquncontrolledrandom walken_US
dc.titleExit Problem and Stochastic Resonance for a Class of Random Perturbationsen_US
dc.typeDissertationen_US

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