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dc.contributor.authorAlmada Monter, Sergio Angelen_US
dc.date.accessioned2011-07-06T16:25:08Z
dc.date.available2011-07-06T16:25:08Z
dc.date.issued2011-04-01en_US
dc.identifier.urihttp://hdl.handle.net/1853/39518
dc.description.abstractA stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.en_US
dc.publisherGeorgia Institute of Technologyen_US
dc.subjectStochastic calculusen_US
dc.subjectSmall noiseen_US
dc.subjectStochastic dynamicsen_US
dc.subjectProbabilityen_US
dc.subjectDynamical systemsen_US
dc.subject.lcshStochastic differential equations
dc.subject.lcshStochastic analysis
dc.subject.lcshDynamics
dc.titleScaling limit for the diffusion exit problemen_US
dc.typeDissertationen_US
dc.description.degreePh.D.en_US
dc.contributor.departmentMathematicsen_US
dc.description.advisorCommittee Chair: Bakhtin, Yuri; Committee Member: Bunimovich, Leonid; Committee Member: Cvitanovic, Pedrag; Committee Member: Houdre, Christian; Committee Member: Koltchinskii, Vladimir; Committee Member: Swiech, Andrzejen_US


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