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We consider a system of $d$ non-linear stochastic heat equations in spatialdimension $k \geq 1$, whose solution is an $\R^d$-valued random field $u=\{u(t\,,x),\, (t,x) \in \R_+ \times \R^k\}$. The $d$-dimensional driving noiseis white in time and with a spatially homogeneous covariance defined as a Rieszkernel with exponent $\beta$, where $0<\beta<(2 \wedge k)$. The non-linearitiesappear both as additive drift terms and as multipliers of the noise. Usingtechniques of Malliavin calculus, we establish an upper bound on the two-pointdensity, with respect to Lebesgue measure, of the $\R^{2d}$-valued randomvector $(u(s,y),u(t,x))$, that, in particular, quantifies how this densitydegenerates as $(s,y) \to (t,x)$. From this result, we deduce a lower bound onhitting probabilities of the process $u$, in terms of Newtonian capacity. Wealso establish an upper bound on hitting probabilities of the process in termsof Hausdorff measure. These estimates make it possible to show that points arepolar when $d > \frac{4+2k}{2-\beta}$ and are not polar when $d <\frac{4+2k}{2-\beta}$. %(if $\frac{4+2k}{2-\beta}$ is an integer, then the case$d=\frac{4+2k}{2-\beta}$ is open). In the first case, we also show that theHausdorff dimension of the range of the process is $\frac{4+2k}{2-\beta}$ a.s.

Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension $k\geq 1$