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We consider a system of $d$ non-linear stochastic fractional heat equationsin spatial dimension $1$ driven by multiplicative $d$-dimensional space-timewhite noise. We establish a sharp Gaussian-type upper bound on the two-pointprobability density function of $(u(s, y), u (t, x))$. From this result, wededuce 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 termsof Newtonian capacity, which is as sharp as that in the Gaussian case. Thisalso improves the result in Dalang, Khoshnevisan and Nualart [\textit{Probab.Theory Related Fields} \textbf{144} (2009) 371--424] for systems of classicalstochastic heat equations. We also establish upper bounds on hittingprobabilities of the solution in terms of Hausdorff measure.

Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations