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Boundary crossing identities for Brownian motion and some nonlinear ode's

Authors: L. Alili and P. Patie
Journal: Proc. Amer. Math. Soc. 142 (2014), 3811-3824
MSC (2010): Primary 35K05, 60J50, 60J65
Published electronically: August 14, 2014
MathSciNet review: 3251722
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Abstract: We start by introducing a nonlinear involution operator which maps the space of solutions of Sturm-Liouville equations into the space of solutions of the associated equations which turn out to be nonlinear ordinary differential equations. We study some algebraic and analytical properties of this involution operator as well as some properties of a two-parameter family of operators describing the set of solutions of Sturm-Liouville equations. Next, we show how a specific composition of these mappings allows us to connect, by means of a simple analytical expression, the law of the first passage time of a Brownian motion over a curve to a two-parameter family of curves. We offer three different proofs of this fact which may be of independent interests. In particular, one is based on the construction of parametric time-space harmonic transforms of the law of some Gauss-Markov processes. Another one, which is of algebraic nature, relies on the Lie group symmetry methods applied to the heat equation and reveals that our two-parameter transformation is the unique nontrivial one.

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Additional Information

L. Alili
Affiliation: Department of Statistics, The University of Warwick, Coventry CV4 7AL, United Kingdom

P. Patie
Affiliation: School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853.

Received by editor(s): November 9, 2012
Published electronically: August 14, 2014
Additional Notes: This work was partially supported by the Actions de Recherche Concertées IAPAS, a fund of the Communautée française de Belgique. The first author is greatly indebted to the Agence Nationale de la Recherche for the research grant ANR-09-Blan-0084-01. The second author would like to thank P. Lescot for several discussions on the Lie group method.
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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