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Scaling limits for conditional diffusion exit problems and asymptotics for nonlinear elliptic equations


Authors: Yuri Bakhtin and Andrzej Święch
Journal: Trans. Amer. Math. Soc. 368 (2016), 6487-6517
MSC (2010): Primary 60J60, 35J15, 35F21
DOI: https://doi.org/10.1090/tran/6574
Published electronically: December 22, 2015
MathSciNet review: 3461040
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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 type. Namely, 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 one to reduce the problem to the Levinson case. We devote an appendix to a rigorous and general discussion of $ h$-transform.


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Additional Information

Yuri Bakhtin
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012

Andrzej Święch
Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160

DOI: https://doi.org/10.1090/tran/6574
Keywords: Diffusion, exit problems, scaling limit, small noise, Doob's $h$-transform, Hamilton--Jacobi--Bellman equation, elliptic PDE, viscosity solution, region of strong regularity.
Received by editor(s): October 22, 2013
Received by editor(s) in revised form: August 25, 2014
Published electronically: December 22, 2015
Additional Notes: The first author was partially supported by NSF via DMS-1407497 and CAREER DMS-0742424. The second author was partially supported by NSF grant DMS-0856485.
Article copyright: © Copyright 2015 American Mathematical Society