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July 2004 Potential theory for hyperbolic SPDEs
Robert C. Dalang, Eulalia Nualart
Ann. Probab. 32(3): 2099-2148 (July 2004). DOI: 10.1214/009117904000000685

Abstract

We give general sufficient conditions which imply upper and lower bounds for the probability that a multiparameter process hits a given set E in terms of a capacity of E related to the process. This extends a result of Khoshnevisan and Shi [Ann. Probab. 27 (1999) 1135–1159], where estimates for the hitting probabilities of the (N,d) Brownian sheet in terms of the (d2N) Newtonian capacity are obtained, and readily applies to a wide class of Gaussian processes. Using Malliavin calculus and, in particular, a result of Kohatsu-Higa [Probab. Theory Related Fields 126 (2003) 421–457], we apply these general results to the solution of a system of d nonlinear hyperbolic stochastic partial differential equations with two variables. We show that under standard hypotheses on the coefficients, the hitting probabilities of this solution are bounded above and below by constants times the (d4) Newtonian capacity. As a consequence, we characterize polar sets for this process and prove that the Hausdorff dimension of its range is min (d,4) a.s.

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Robert C. Dalang. Eulalia Nualart. "Potential theory for hyperbolic SPDEs." Ann. Probab. 32 (3) 2099 - 2148, July 2004. https://doi.org/10.1214/009117904000000685

Information

Published: July 2004
First available in Project Euclid: 14 July 2004

zbMATH: 1054.60066
MathSciNet: MR2073187
Digital Object Identifier: 10.1214/009117904000000685

Subjects:
Primary: 60H15 , 60J45
Secondary: 60G60 , 60H07

Keywords: Gaussian process , hitting probability , multiparameter process , nonlinear hyperbolic SPDE , potential theory

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3 • July 2004
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