Scaling limits for conditional diffusion exit problems, Doob's h-transform, and asymptotics for nonlinear elliptic equations

Yury Bakhtin, Andrzej Swiech

Research output: Contribution to journalArticle

Abstract

The goal of this paper is to supplement the large deviation principle of the Freidlin--Wentzell theory on exit problems for diffusion processes with results of classical central limit theorem kind. We describe a class of situations where conditioning on exit through unlikely locations leads to a Gaussian scaling limit for the exit distribution. Our results are based on Doob's h-transform and new asymptotic convergence gradient estimates for elliptic nonlinear equations that allow to reduce the problem to the Levinson case. We devote a separate section to a rigorous and general discussion of h-transform.
Original languageEnglish (US)
Article number1310.6023
JournalarXiv
StatePublished - Oct 22 2013

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H-transform
Exit Problem
Nonlinear Elliptic Equations
Scaling Limit
Diffusion Problem
Convergence Estimates
Gradient Estimate
Asymptotic Convergence
Large Deviation Principle
Conditioning
Central limit theorem
Diffusion Process

Keywords

  • math.PR
  • math.AP
  • math.DS
  • 60J60, 35J15, 35F21

Cite this

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title = "Scaling limits for conditional diffusion exit problems, Doob's h-transform, and asymptotics for nonlinear elliptic equations",
abstract = "The goal of this paper is to supplement the large deviation principle of the Freidlin--Wentzell theory on exit problems for diffusion processes with results of classical central limit theorem kind. We describe a class of situations where conditioning on exit through unlikely locations leads to a Gaussian scaling limit for the exit distribution. Our results are based on Doob's h-transform and new asymptotic convergence gradient estimates for elliptic nonlinear equations that allow to reduce the problem to the Levinson case. We devote a separate section to a rigorous and general discussion of h-transform.",
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