Scaling limit for the diffusion exit problem

dc.contributor.advisor Bakhtin, Yuri
dc.contributor.author Almada Monter, Sergio Angel en_US
dc.contributor.committeeMember Bunimovich, Leonid
dc.contributor.committeeMember Cvitanovic, Pedrag
dc.contributor.committeeMember Houdre, Christian
dc.contributor.committeeMember Koltchinskii, Vladimir
dc.contributor.department Mathematics en_US
dc.date.accessioned 2011-07-06T16:25:08Z
dc.date.available 2011-07-06T16:25:08Z
dc.date.issued 2011-04-01 en_US
dc.description.abstract A 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.description.degree Ph.D. en_US
dc.identifier.uri http://hdl.handle.net/1853/39518
dc.publisher Georgia Institute of Technology en_US
dc.subject Stochastic calculus en_US
dc.subject Small noise en_US
dc.subject Stochastic dynamics en_US
dc.subject Probability en_US
dc.subject Dynamical systems en_US
dc.subject.lcsh Stochastic differential equations
dc.subject.lcsh Stochastic analysis
dc.subject.lcsh Dynamics
dc.title Scaling limit for the diffusion exit problem en_US
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
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