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

 

Second-order hyperbolic s.p.d.e.'s driven by homogeneous Gaussian noise on a hyperplane


Authors: Robert C. Dalang and Olivier Lévêque
Journal: Trans. Amer. Math. Soc. 358 (2006), 2123-2159
MSC (2000): Primary 60H15; Secondary 60G15, 35R60
Published electronically: May 9, 2005
MathSciNet review: 2197451
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Abstract: We study a class of hyperbolic stochastic partial differential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spatially homogeneous within the hyperplane. Two natural notions of solutions are function-valued solutions and random field solutions. For the linear form of the equations, we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution, and it turns out that the conditions differ. In spatial dimensions 2 and 3, under the condition for existence of a random field solution to the linear form of the equation, we prove existence and uniqueness of a random field solution to non-linear forms of the equation.


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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

Olivier Lévêque
Affiliation: Institut de Systèmes de Communication, Ecole Polytechnique Fédérale, Station 14, 1015 Lausanne, Switzerland
Email: olivier.leveque@epfl.ch

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03740-2
PII: S 0002-9947(05)03740-2
Keywords: Stochastic partial differential equations, spatially homogeneous Gaussian noise, hyperbolic equations
Received by editor(s): January 26, 2004
Received by editor(s) in revised form: May 4, 2004
Published electronically: May 9, 2005
Additional Notes: The research of the first author was partially supported by the Swiss National Foundation for Scientific Research
This article is based on part of the second author’s Ph.D. thesis, written under the supervision of the first author.
Article copyright: © Copyright 2005 American Mathematical Society