Abstract
Let $(X_t)_{t \geq 0}$ denote the measure-valued critical branching Brownian motion. When the support of the initial state, $X_0$, is bounded, temporally global results are given concerning the range, i.e., the size of the supports of $(X_t)_{t \geq 0}$, and the hitting (i.e., charging) probabilities of distant balls are evaluated asymptotically; they depend strongly on the dimension, $d$, of the underlying Euclidean space $\mathbb{R}^d$. In contrast, in the case $d = 1$ and $X_0 = \lambda$ (Lebesgue measure), it is shown that (spatially) local extinction occurs. Also extensions are indicated for the case of an infinite variance branching mechanism; these results are also dimensionally dependent.
Citation
I. Iscoe. "On the Supports of Measure-Valued Critical Branching Brownian Motion." Ann. Probab. 16 (1) 200 - 221, January, 1988. https://doi.org/10.1214/aop/1176991895
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