Criteria for hitting probabilities with applications to systems of stochastic wave equations. (English) Zbl 1218.60054

Dalang, Robert C.; Sanz-Solé, Marta

Bernoulli 16, No. 4, 1343-1368 (2010).

This work develops some very important and general results concerning the hitting probabilities for a general \({\mathbb R}^{d}\)-valued stochastic process \(V=\{v(x), x \in{\mathbb R}^{m}\}\) where \(v\) is a random map. The hitting probability, defined as \(P(v(I)\cap A \neq \emptyset)\), for a fixed compact set \(I \subset {\mathbb R}^{m}\) (of positive Lebesgue measure) and an arbitrary Borel set \(A \subset {\mathbb R}^{d}\), is estimated in terms of the capacity and the Hausdorff measure of \(A\), using estimates which highlight the role of the spatial dimension \(m\) and of course \(d\). These results are of interests in their own right as they offer an insight into the effects of dimensionality on the hitting probabilities for very general classes of stochastic processes.
Then, special attention is paid to Gaussian processes, and refinements of the general results are provided, as well as conditions on a Gaussian process \(\{X(t)\}\) such that the density function of \((X(t),X(s))\) is bounded above. The general results are finally applied to the case where the stochastic process is a solution of a system of \(d\) stochastic wave equations in spatial dimensions \(m \geq 1\) to provide bounds of the form
\[ \text{Cap}_{d-2(m+1)/(2-\beta)}(A)\leq P(u([t_{0},T]\times [-M,M]^m)\cap A \neq \emptyset)\leq C \, {\mathcal H}_{d-2(m+1)/(2-\beta)}(A) \]
where Cap\(_{\gamma}(A)\), \({\mathcal H}_{\gamma}(A)\) are the capacity and the Hausdorff dimension of the set \(A\), respectively. Such bounds provide important and useful qualitative information on the solution, which, so far, has not been studied to this extent for the wave equation, on account of the difficulties associated with the integral representation of their solution due to the singularities of the integral kernel.

Reviewer: Athanasios Yannacopoulos (Athens)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations

Keywords:

capacity; Hausdorff measure; hitting probabilities; spatially homogeneous colored noise; systems of stochastic wave equations

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