Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension $k \geq 1$

Abstract

We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix $\gamma _{Z}$ of $Z := (u(s, y), u(t, x) - u(s, y))$, where $u$ is the solution to a system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$. We also obtain the optimal exponents for the $L^{p}$-modulus of continuity of the increments of the solution and of its Malliavin derivatives. These lead to optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty [ \times \mathbb{R} ^{k}\}$ in the non-Gaussian case in terms of Newtonian capacity, and improve a result in Dalang, Khoshnevisan and Nualart [Stoch PDE: Anal Comp 1 (2013) 94–151].