Hitting probabilities for systems of stochastic partial differential equations: an overview

We consider a d-dimensional random field that solves a possibly non-linear system of stochastic partial differential equations, such as stochastic heat or wave equations. We present results on upper and lower bounds for the probabilities that the random field visits a deterministic subset of $\mathbb{R}^d$, in terms, respectively, of Hausdorff measure and Newtonian capacity of the subset. These bounds determine the critical dimension above which points are polar, but do not, in general, determine whether points are polar in the critical dimension. For linear spde's, we resolve, in joint work with Carl Mueller and Yimin Xiao, the issue of polarity of points in the critical dimension, and also address the question of existence of multiple points in critical dimensions