Abstract: Let $X^H=\{X^H(s), s\in \mathbb{R}^{N_1}\} $ and $X^K=\{X^K(t), t\in \mathbb{R}^{N_2}\} $ be two independent time-space anisotropic random fields with indices $H \in (0,1)^{N_1}$ and $K \in (0,1)^{N_2}$, which may not possess Gaussianity, and which take values in $\mathbb{R}^d$ with a space metric $\tau$. Under certain general conditions with density functions defined on a bounded interval, we study problems regarding the hitting probabilities of time-space anisotropic random fields and the existence of intersections of the sample paths of random fields $X^H$ and $X^K$. More generally, for any Borel set $F \subset \mathbb{R}^d$, the conditions required for $F$ to contain intersection points of $X^H$ and $X^K$ are established. As an application, we give an example of an anisotropic non-Gaussian random field to show that these results are applicable to the solutions of non-linear systems of stochastic fractional heat equations. }

Key words: Hitting probability, multiple intersection, anisotropic random field, capacity, Hausdorff dimension, stochastic fractional heat equations