2008/11/01 by Olav Kallenberg · 1 citation
Mathematics · Physics and Astronomy · #Hitting time #Lebesgue integration #Lebesgue measure #Limiting #Mathematical proof #Measure (data warehouse) #Point (geometry) #Random Matrices and Applications #Random measure #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60G57 #msc:60J60 #msc:60J80
paper · pdf · doi:10.1214/07-aop386
published as Annals of Probability 2008, Vol. 36, No. 6, 2176-2214 · Published in at http://dx.doi.org/10.1214/07-AOP386 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2008/11/01 · arxiv created 2009/01/19 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
Let ξ be a Dawson–Watanabe superprocess in ℝd such that ξt is a.s. locally finite for every t≥0. Then for d≥2 and fixed t>0, the singular random measure ξt can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the ɛ-neighborhoods of supp ξt. When d≥3, the local distributions of ξt near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure ξ̃. By contrast, the corresponding distributions for d=2 are locally invariant. Further results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities. Our main proofs are based on a detailed analysis of the historical structure of ξ.