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

Abstract

We consider a system of \(d\) non-linear stochastic heat equations in spatial dimension \(k \ge 1\), whose solution is an \(\mathbb{R }^d\)-valued random field \(u= \{u(t\,,x),\, (t,x) \in \mathbb{R }_+ \times \mathbb{R }^k\}\). The \(d\)-dimensional driving noise is white in time and with a spatially homogeneous covariance defined as a Riesz kernel with exponent \(\beta \), where \(0 \frac{4+2k}{2-\beta }\) and are not polar when \(d< \frac{4+2k}{2-\beta }\). In the first case, we also show that the Hausdorff dimension of the range of the process is \(\frac{4+2k}{2-\beta }\) a.s.