The Annals of Probability

On the Large Time Growth Rate of the Support of Supercritical Super- Brownian Motion

Ross G. Pinsky

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Abstract

Consider the supercritical super-Brownian motion $X(t,\cdot)$ on $R^d$ corresponding to the evolution equation $u_t = \frac{D}{2}\Delta u + u - u^2.$ We obtain rather tight bounds on $P_\mu(X(s,B^c_n(0)) = 0$, for all $s \in \lbrack 0,t\rbrack)$ and on $P_\mu(X(t,B^c_n(0)) = 0)$, for large $n$, where $P_\mu$ denotes the measure corresponding to the supercritical super-Brownian motion starting from the finite measure, $\mu, B_n(0) \subset R^d$ denotes the ball of radius $n$ centered at the origin and $B^c_n(0)$ denotes its complement. In particular, we show, for example, that if $\mu$ is a compactly supported, finite measure on $R^d$, then $\lim_{n\rightarrow\infty} P_\mu(X(t,B^c_n(0)) = 0, \text{for all} t \in \lbrack 0,\gamma n\rbrack) = 1 \text{if} \gamma < (2D)^{-1/2}$ and $\lim_{n\rightarrow\infty} P_\mu(X(\gamma n, B^c_n(0)) = 0\mid\text{the process survives}) = 0 \text{if} \gamma > (2D)^{-1/2}.$

Article information

Source
Ann. Probab., Volume 23, Number 4 (1995), 1748-1754.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176987801

Digital Object Identifier
doi:10.1214/aop/1176987801

Mathematical Reviews number (MathSciNet)
MR1379166

Zentralblatt MATH identifier
0852.60094

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J25: Continuous-time Markov processes on general state spaces

Keywords
Super-Brownian motion supercritical measure-valued process

Citation

Pinsky, Ross G. On the Large Time Growth Rate of the Support of Supercritical Super- Brownian Motion. Ann. Probab. 23 (1995), no. 4, 1748--1754. doi:10.1214/aop/1176987801. https://projecteuclid.org/euclid.aop/1176987801


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