A Strong Law of Large Numbers for Super-Critical Branching Brownian Motion with Absorption
- PDF / 508,782 Bytes
- 26 Pages / 439.37 x 666.142 pts Page_size
- 54 Downloads / 171 Views
A Strong Law of Large Numbers for Super-Critical Branching Brownian Motion with Absorption Oren Louidor1 · Santiago Saglietti1,2 Received: 20 February 2020 / Accepted: 31 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two moments, and in which all particles have a drift towards the origin where they are immediately absorbed. It is well-known that the population survives forever with positive probability if and only if the branching rate is sufficiently large. We show that throughout this super-critical regime, the number of particles inside any fixed set normalized by the mean population size converges to an explicit limit, almost surely and in L 1 . As a consequence, we get that almost surely on the event of eternal survival, the empirical distribution of particles converges weakly to the (minimal) quasi-stationary distribution associated with the Markovian motion driving the particles. This proves a result of Kesten (Stoch Process Appl 7(1):9–47, 1978) from 1978, for which no proof was available until now.
1 Introduction and Results Given some fixed c > 0, let X = (X t )t≥0 be a Brownian motion with drift −c and variance coefficient 1, which is absorbed upon reaching the origin, i.e. X is the process given by X t := X 0 − c(t ∧ H0 ) + Wt∧H0
(1)
for each t, where W is a standard Brownian motion on R and H0 := inf{s ≥ 0 : X 0 − cs + Ws = 0}. Now, consider the following branching dynamics associated with X : i. The dynamics starts with a single particle, located initially at some x ≥ 0, whose position evolves randomly according to X .
Communicated by Alessandro Giuliani.
B
Santiago Saglietti [email protected] Oren Louidor [email protected]
1
Technion, Haifa, Israel
2
Pontificia Universidad Católica de Chile, Santiago, Chile
123
O. Louidor, S. Saglietti
ii. This initial particle branches at a fixed rate r > 0 (independently of the motion it describes) and, whenever it does so, it dies and gets replaced at its current position by an independent random number of particles m having some fixed distribution μ on N0 . iii. Starting from their birth position, now each of these m new particles independently mimics the same stochastic behavior of its parent. iv. If a particle has 0 children, then it simply dies and disappears from the dynamics. We will call this the (c, r , μ)-branching dynamics associated with X (or simply (c, r , μ)dynamics). Let us agree on the following notation to be used throughout the sequel: • For each t ≥ 0 we denote by At the collection of all particles present in the dynamics at time t. • For any particle u ∈ At and 0 ≤ s ≤ t we let u s be the position of the unique ancestor of u (including u itself) which belongs to As . Furthermore, we will write u t := (u s )s∈[0,t] to denote its trajectory in the time interval [0, t]. • We will write B(0,+∞) for the class of all Borel subsets of (0, +∞) and, for any given t ≥ 0 a
Data Loading...
