We consider a system of d coupled non linear stochastic heat equations in spatial dimension 1, driven by d-dimensional space-time white noise. The solution of this system is a process indexed by space-time, with values in R^d. The main objective is to determine, for a given subset of R^d, whether or not this set is hit by the space-time process. Using Malliavin calculus, we obtain in [1] and [2] upper and lower bounds on the univariate and bivariate joint densities of the solution. This leads to upper and lower bounds on hitting probabilities for the space-time process, in terms of capacity and Hausdorff measure of the sets. We also obtain related estimates when one of the space-time parameters is fixed. This makes it possible to determine the critical dimension above which points are polar, as well as the Hausdorff dimensions of the range of the process and of its level sets.

1. Dalang, R.C., Khoshnevisan, D., Nualart, E., "Hitting probabilities for the non-linear stochastic heat equation with additive noise" (preprint, 2007; http://arxiv.org/PS_cache/math/pdf/0702/0702710v1.pdf).
2. Dalang, R.C., Khoshnevisan, D., Nualart, E., "Hitting probabilities for the non-linear stochastic heat equation with multiplicative noise" (preprint, 2007; http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.1312v1.pdf).