Branching Brownian motion with an inhomogeneous breeding potential
A driftless branching Brownian motion (BBM) is considered: each particle splits into two particles at rate \(\beta |y|^p, \beta > 0, p \in [0,2]\) (the breeding potential), where \(y \in \mathbb R\) is the particle's spatial position. Denote by \(Y_u(t)\) the spatial position at time \(t\) of particle \(u \in N_t\) the set of particles alive at time \(t\) and let \(P^x\) be the probability law of the process starting from a single particle located at the point \(x \in \mathbb R\). For the breeding potential with \(p > 2\) the population would \(P^x\)-almost surely explode in a finite time [cf. \textit{K. Itô} and \textit{H. P. McKean}, Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer-Verlag (1965; Zbl 0127.09503)]. The authors define the \textit{right-most particle} of the BBM to be the supremum of spatial positions of the particles from \(N_t\) and establish the following asymptotics for \(R_t\) as \( t \to \infty\): - if \( p \in [0,2)\), then \(R_t \sim a t^{2/(2-p)}\) (with \(a\) given by an explicit formula involving \(t\) only, \(a \to \infty\) when \(p \to 2-\)), - if \(p = 2\), then \(R_t \sim \exp {(t \sqrt {2\beta})}\). (The authors remark in the abstract that the above result for the constant breeding, \(p = 0\), ``is long established''.) Proofs rest on analysis of certain additive martingales (stemming from the BBM) and related spine changes of measure. The notions and facts needed are supplied in two separate sections preceding the proof.
scientific article; zbMATH DE number 5611489
scientific article