Limiting Distribution of Particles Near the Frontier in the Catalytic Branching Brownian Motion
- PDF / 850,198 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 96 Downloads / 197 Views
Limiting Distribution of Particles Near the Frontier in the Catalytic Branching Brownian Motion Sergey Bocharov1
Received: 28 March 2019 / Accepted: 27 December 2019 © Springer Nature B.V. 2020
Abstract We consider the model of branching Brownian motion with a single catalytic point at the origin and binary branching. We establish some fine results for the asymptotic behaviour of the numbers of particles travelling at different speeds and in particular prove that the point process of particles travelling at the critical speed converges in distribution to a mixture of Poisson point processes. Keywords Branching Brownian motion · Catalytic branching Mathematics Subject Classification 60J80 · 60F15
1 Introduction and Main Results 1.1 Description of the Model Branching Brownian motion with a single-point catalyst at the origin is a spatial population model in which individuals (referred to as particles) move in space according to the law of standard Brownian motion and reproduce themselves at a spatially-inhomogeneous branching rate βδ0 (·), where δ0 (·) is the Dirac delta measure and β > 0 is some constant. More precisely, in such a process we start with a single particle at some initial location x0 ∈ R at time 0 whose position Xt at time t ≥ 0 up until the time it dies evolves like a standard Brownian motion. At a random time T satisfying P x0 T > t | (Xs )s≥0 = e−βLt , The author is supported by NSFC grant (No. 11731012) and the Fundamental Research Funds for the Central Universities.
B S. Bocharov
[email protected]
1
School of Mathematical Sciences, Zhejiang University, No. 38 Zheda Rd., Hangzhou, Zhejiang 31002, P.R. China
S. Bocharov
where (Lt )t≥0 is the local time at 0 of (Xt )t≥0 , the initial particle dies and is replaced with two new particles, which independently of each other and of the previous history stochastically continue the behaviour of their parent starting from time T and position XT = 0. That is, they move like Brownian motions, die after random times giving birth to two new particles each, etc. t Note that informally we may write Lt = 0 δ0 (Xs )ds thus justifying calling the branching rate βδ0 (·). This is made precise by the theory of additive functionals of Brownian motion. See, for example, papers of Chen and Shiozawa [13] and Shiozawa [23–25] where they study a large class of processes with branching rates which are allowed to be measures. Let us mention that in the past catalytic branching processes have also been studied in the context of superprocesses (see for example papers of Dawson and Fleischmann [14], Engländer and Turaev [15] and Vatutin and Xiong [26]) and also in the context of branching random walks on integer lattices, both in discrete time (see, for example, a paper of Carmona and Hu [12]) and continuous time (see, for example, a paper of Bulinskaya [10]). Also, a closely related type of processes is branching Brownian motions with the branching rate given by either a compactly-supported function or a function decaying sufficiently fast at infinity (see, for e
Data Loading...
