Survival Asymptotics for Branching Brownian Motion in a Poissonian Trap Field,

J. Englander, F. den Hollander

2003, v.9, №3, 363-389

ABSTRACT

In this paper we study a branching Brownian motion on $\r^d$ with branching rate $\beta$ in a Poissonian field of spherical traps whose locally finite intensity measure $\nu$ is such that $\mathrm{d}\nu/\mathrm{d}x$ decays radially with the distance to the origin as $\mathrm{d}\nu/\mathrm{d}x \sim\ell/|x|^{d-1}$, $|x|\to\infty$. The process starts with a single particle at the origin. The annealed probability that none of the particles hits a trap until time $t$ is shown to decay like $\exp[-I(\ell,\beta,d)t+o(t)]$ as $t\to\infty$, where the rate constant $I(\ell,\beta,d)$ is computed in terms of a variational problem. It turns out that this rate constant exhibits a crossover at a critical value $\ell_{\mathit cr}=\ell_{\mathit cr}(\beta,d)$. We prove that, conditional on survival until time $t$, the following properties hold with a probability tending to one as $t\to\infty$. For $\ell<\ell_{\mathit cr}$, a ball of radius $\sqrt{2\beta}\,t$ around the origin is emptied, the branching particle system stays inside this ball and branches at rate $\beta$. For $\ell>\ell_{\mathit cr}$, on the other hand, the system \begin{itemize} \item[$d=1$:] suppresses the branching until time $t$, empties a ball of radius $o(t)$ around the origin (i.e., a ball whose radius is larger than the trap radius but smaller than order $t$), and stays inside this ball; \item[$d\geq 2$:] suppresses the branching until time $\eta^*t$, empties a ball of radius $\vphantom{T^{\sum}}\sqrt{2\beta}\,(1-\eta^*)t$ around a point at distance $c^*t$ from the origin, and during the remaining time $(1-\eta^*)t$ branches at rate $\beta$. Here, $0<\eta^*<1$ and $c^*>0$ are the minimizers of the variational problem for $I(\ell,\beta,d)$. \end{itemize} In the latter case, we show that one optimal survival strategy is the following: the system completely suppresses the branching until time $\eta^*t$, i.e., only the initial particle is alive at time $\eta^*t$, within a small empty tube moves the initial particle to a point at distance $c^*t$ from the origin, empties a ball of radius $\sqrt{2\beta}\,(1-\eta^*)t$ around that point, stays inside this ball during the remaining time $(1-\eta^*)t$ and branches at rate $\beta$. There are, however, {\it other} survival strategies with the same exponential cost, which cannot be distinguished without a higher-order analysis. Remarkably, it turns out that $\eta^*$ and $c^*$ tend to a strictly positive limit as $\ell\downarrow\ell_{\mathit{cr}}$, i.e., the crossover at $\ell_{\mathit{cr}}$ is discontinuous. Moreover, $c^*>\sqrt{2\beta}\,(1-\eta^*)$ for all $\ell>\ell_{\mathit{cr}}$, i.e., the empty ball does not contain the origin. In contrast, the annealed probability that at least one of the particles does not hit a trap up to time $t$ is shown to decay to a strictly positive limit.

Keywords: branching Brownian motion,Poissonian traps,large deviations,survival probability,variational problem,optimal survival strategy

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