Path properties of the solution to the stochastic heat equation with L\'evy noise

We consider sample path properties of the solution to the stochastic heat equation, in ${\mathbb{R}}^d$ or bounded domains of ${\mathbb{R}}^d$, driven by a L'evy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a c`adl`ag modification in fractional Sobolev spaces of index less than $-\frac d 2$. Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal-Getoor index of the L'evy noise such that noises with a smaller index entail continuous sample paths, while L'evy noises with a larger index entail sample paths that are unbounded on any non-empty open subset. Our results apply to additive as well as multiplicative L'evy noises, and to light- as well as heavy-tailed jumps.