Exceptional times of compact support for $α$-stable superprocesses

Abstract

We consider symmetric $α$-stable Dawson--Watanabe superprocesses on $\mathbb R^d$ for every $d\geq1$ and $0<α<2$, started from an arbitrary nonzero finite measure. We prove that, almost surely, the times at which the spatial support is compact and nonempty have Hausdorff dimension one in every nontrivial interval before extinction. We also prove that, almost surely, the times at which the support is contained in any prescribed neighborhood of the extinction point have Hausdorff dimension one in every nontrivial interval ending at extinction. These results extend the corresponding conclusions for $0<α<2/3$ to the full stable range. The key new ingredient is a uniform zero-set theorem of independent interest for Feller branching diffusions started from zero with deterministic continuous cumulative immigration. It shows that sufficiently small total immigration ensures that the zero set on a prescribed open interval has Hausdorff dimension arbitrarily close to one, with arbitrarily high probability. The estimate is uniform over the immigration profile and allows singular continuous immigration measures.