Exceptional times for the instantaneous propagation of superprocess

TL;DR

This paper investigates the support properties of superprocesses with symmetric stable spatial motion on the real line, revealing the existence of dense sets of exceptional times with compact support and extending previous results to a broader range of stability indices.

Contribution

It demonstrates the existence of dense exceptional times with compact support for superprocesses with stable index  /3, extending prior work from   to  /3.

Findings

Existence of dense sets of exceptional times with compact support.

Support concentrates near extinction points at certain exceptional times.

Full Hausdorff dimension of the set of exceptional times.

Abstract

For a Dawson-Watanabe superprocess $X$ on $\mathbb{R}^d$, it is shown in Perkins (1990) that if the underlying spatial motion belongs to a certain class of L\'evy processes that admit jumps, then with probability one the closed support of $X_t$ is the whole space for almost all $t>0$ before extinction, the so-called ``instantaneous propagation'' property. In this paper for superprocesses on $\mathbb{R}^1$ whose spatial motion is the symmetric stable process of index $\alpha \in (0,2/3)$, we prove that there exist exceptional times at which the support is compact and nonempty. Moreover, we show that the set of exceptional times is dense with full Hausdorff dimension. Besides, we prove that near extinction, the support of the superprocess is concentrated arbitrarily close to the distinction point, thus upgrading the corresponding results in Tribe (1992) from $\alpha \in (0,1/2)$ to $\alpha \in (0,2/3)$, and we further show that the set of such exceptional times also admits a full Hausdorff dimension.