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

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 $\lim_{n\rightarrow\infty} P_\mu(X(\gamma n, B^c_n(0)) = 0\mid\text{the process survives}) = 0 \text{if} \gamma > (2D)^{-1/2}.$