The Annals of Probability

Potential theory for hyperbolic SPDEs

Robert C. Dalang and Eulalia Nualart

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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

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

First available in Project Euclid: 14 July 2004

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Zentralblatt MATH identifier

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

Potential theory nonlinear hyperbolic SPDE multiparameter process Gaussian process hitting probability


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

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