The Annals of Probability

Potential theory for hyperbolic SPDEs

Robert C. Dalang and Eulalia Nualart

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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.

Article information

Source
Ann. Probab., Volume 32, Number 3 (2004), 2099-2148.

Dates
First available in Project Euclid: 14 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1089808421

Digital Object Identifier
doi:10.1214/009117904000000685

Mathematical Reviews number (MathSciNet)
MR2073187

Zentralblatt MATH identifier
1054.60066

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60G60: Random fields

Keywords
Potential theory nonlinear hyperbolic SPDE multiparameter process Gaussian process hitting probability

Citation

Dalang, Robert C.; Nualart, Eulalia. Potential theory for hyperbolic SPDEs. Ann. Probab. 32 (2004), no. 3, 2099--2148. doi:10.1214/009117904000000685. https://projecteuclid.org/euclid.aop/1089808421


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