Potential theory for hyperbolic SPDEs
成果类型：
Article
署名作者：
Dalang, RC; Nualart, E
署名单位：
Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/009117904000000685
发表日期：
2004
页码：
2099-2148
关键词：
摘要：
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 (d - 2N) 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 (d - 4) 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.
来源URL：