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

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$u$isthesolutiontosystemof$d$ 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} $\mathbf{\text{1}}$ (2013) 94–151].