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

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 system of $d$ non-linear stochastic heat equations in spatial dimension $k\ge 1$. 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 $\{u(t,x):(t,x)\in [0,\infty [\times \mathbb{R}\}$ in the non-Gaussian case in terms of Newtonian capacity, and improve a result in Dalang, Khoshnevisan and Nualart [\textit{Stoch PDE: Anal Comp} \textbf{1} (2013) 94--151].

34 pages