Potential theory for hyperbolic SPDEs

By Robert C. Dalang, E. Nualart. 2004

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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 (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.

Subfields: Demography, Finance

View Paper: https://www.semanticscholar.org/paper/750138179821fd9cea1ef01f75bf042422c2e4a0

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Citations in the corpus

Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions (2011, methodology): “This is done following some standard techniques of estimation of Sobolev norms and inverse moments of the determinant of the Malliavin matrix (as in [16] and [6], Section 4), but in our computations we explicitly pop out the dependence with respect to the coefficients of the SDE and their derivative”
Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise (2007, background): “The only potential-theoretic result that we are aware of for systems of non-linear spde’s with multiplicative noise (σ non-constant) is Dalang and Nualart [3], who study the case of the reduced hyperbolic spde on R2+ (essentially equivalent to the wave equation in spatial dimension 1): ∂2 Xit ∂t1∂t2”
Sample Path Properties of Anisotropic Gaussian Random Fields (2009, methodology): “The first method is to use an extension, due to Arnold and Imkeller [2], Funaki, Kikuchi and Potthoff [39], Dalang, Khoshnevisan and Nualart [22], of the powerful Garsia-Rodemich-Rumsey continuity lemma; the second is the “minorizing metric” method of Kwapień and Risiński [60]; and the third is ba”
Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields (2009, methodology): “For the sake of completeness, we will provide a simpler proof by modifying the argument in the proof of Proposition 4.4 in Dalang, Khoshnevisan and Nualart [3].”
Hitting probabilities for systems of non-linear stochastic heat equations with additive noise (2007, background): “If one is only interested in certain of these conclusions, then weaker assumptions than A1 are possible, analogous to Hypothesis H1 of Dalang and Nualart (2004). For instance, Theorem 2.”