For Gaussian random fields with values in Rd, sharp upper and lower bounds on the probability of hitting a fixed set have been available for many years. These apply in particular to the solutions of systems of linear SPDEs. For non-Gaussian random fields, the available bounds are less sharp. For systems of stochastic heat equations, a sharp lower bound was obtained in [1]. Here, we obtain the corresponding sharp upper bound. This is based on joint work with Fei Pu and David Nualart.
