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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.'s


Authors: Robert C. Dalang, Carl Mueller and Roger Tribe
Journal: Trans. Amer. Math. Soc. 360 (2008), 4681-4703
MSC (2000): Primary 60H15; Secondary 60H20
Published electronically: April 14, 2008
MathSciNet review: 2403701
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Abstract: We establish a probabilistic representation for a wide class of linear deterministic p.d.e.'s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution.


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Additional Information

Robert C. Dalang
Affiliation: Institut de Mathématiques, Ecole Polytechnique Fédérale, Station 8, 1015 Lausanne, Switzerland
Email: robert.dalang@epfl.ch

Carl Mueller
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
Email: cmlr@math.rochester.edu

Roger Tribe
Affiliation: Department of Mathematics, University of Warwick, CV4 7AL, United Kingdom
Email: tribe@maths.warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04351-1
PII: S 0002-9947(08)04351-1
Keywords: Feynman-Kac formula, wave equation, probabilistic representation of solutions, stochastic partial differential equations, moment formulae.
Received by editor(s): October 13, 2005
Received by editor(s) in revised form: May 19, 2006
Published electronically: April 14, 2008
Additional Notes: The first author was partially supported by the Swiss National Foundation for Scientific Research
The second author was partially supported by an NSF grant.
Article copyright: © Copyright 2008 American Mathematical Society