Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line

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 nonlinear systems of stochastic heat equations, a sharp lower bound was obtained in a previous paper by two of the authors. Here, we obtain the corresponding sharp upper bound. The proof requires a bound on the joint probability density function of a two-dimensional random vector whose components are the solution to the nonlinear stochastic heat equation and the supremum over a small rectangle of the solution to the linear stochastic heat equation, in terms of the size of the rectangle. This bound makes use of a formula that expresses the density of a locally nondegenerate random vector as an iterated Skorohod integral. The main effort is to estimate, using Malliavin calculus, each of the terms that arise from this formula.