# Deterministic Sets Visited by Random Paths

joint seminar CMAF/GFM, by Marta Sanz-Solé (Univ. Barcelona, Spain)

We are interested in the geometric measure properties of deterministic sets reached by random fields. More specifically, we will analyze conditions which provide upper and lower bounds for hitting probabilities of random fields in terms of the Hausdorff measure and the Bessel-Riesz capacity, respectively. The role of the regularity of the sample paths, and of the properties of probability densities will be highlighted.

As an illustration, we shall consider systems of stochastic wave equations
in spatial dimension *k* > or = to 1. In the non-Gaussian case, *k* will be
restricted to {1; 2; 3}, and we will apply Malliavin calculus. For the sake
of completeness, a brief introduction to these techniques will be presented.
Applications to other examples of stochastic partial differential equations
will be mentioned.

This is joint work with Robert Dalang (EPFL, Switzerland).