2023/11/23 by Jieliang Hong, Hong, Jieliang, Leonid Mytnik +1
Mathematics · #60G17 #60J68 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2311.13757
openalex publication_date 2023/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a Dawson-Watanabe superprocess X on ℝd, it is shown in Perkins (1990) that if the underlying spatial motion belongs to a certain class of Lévy processes that admit jumps, then with probability one the closed support of Xt is the whole space for almost all t>0 before extinction, the so-called ``instantaneous propagation'' property. In this paper for superprocesses on ℝ1 whose spatial motion is the symmetric stable process of index α∈ (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 α∈ (0,1/2) to α∈ (0,2/3), and we further show that the set of such exceptional times also admits a full Hausdorff dimension.