# Some local approximations of Dawson-Watanabe superprocesses - Mathematics > Probability

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Abstract: Let $\xi$ be a Dawson-Watanabe superprocess in $\mathbb{R}^d$ such that$\xi t$ is a.s. locally finite for every $t\geq 0$. Then for $d\geq2$ and fixed$t>0$, the singular random measure $\xi t$ can be a.s. approximated by suitablynormalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoodsof $\operatorname {supp}\xi t$. When $d\geq3$, the local distributions of$\xi t$ near a hitting point can be approximated in total variation by those ofa stationary and self-similar pseudo-random measure $\tilde{\xi}$. By contrast,the corresponding distributions for $d=2$ are locally invariant. Furtherresults include improvements of some classical extinction criteria and somelimiting properties of hitting probabilities. Our main proofs are based on adetailed analysis of the historical structure of $\xi$.

Author: ** Olav Kallenberg**

Source: https://arxiv.org/