A Strong Law of Large Numbers for Super-Critical Branching Brownian Motion with Absorption

@article{Louidor2020ASL,
  title={A Strong Law of Large Numbers for Super-Critical Branching Brownian Motion with Absorption},
  author={Oren Louidor and Santiago Saglietti},
  journal={Journal of Statistical Physics},
  year={2020}
}
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 if and only if the branching rate is sufficiently large, then the population survives forever with positive probability. We show that throughout this super-critical regime, the number of particles inside any fixed set normalized by the mean… 

ON LAWS OF LARGE NUMBERS IN L2 FOR SUPERCRITICAL BRANCHING MARKOV PROCESSES BEYOND λ-POSITIVITY

We give necessary and sufficient conditions for laws of large numbers to hold in L for the empirical measure of a large class of branching Markov processes, including λ-positive systems but also some

On laws of large numbers in $L^{2}$ for supercritical branching Markov processes beyond $\lambda $-positivity

We give necessary and sufficient conditions for laws of large numbers to hold in $L^2$ for the empirical measure of a large class of branching Markov processes, including $\lambda$-positive systems

Maximal displacement and population growth for branching Brownian motions

We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schr\"odinger type operator. We first

References

SHOWING 1-10 OF 35 REFERENCES

Survival of Near-Critical Branching Brownian Motion

Consider a system of particles performing branching Brownian motion with negative drift $\mu= \sqrt{2 - \varepsilon}$ and killed upon hitting zero. Initially there is one particle at x>0. Kesten

Critical branching Brownian motion with absorption: survival probability

We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $$-\sqrt{2}$$-2. Kesten

The extremal process of branching Brownian motion

AbstractWe prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson

The genealogy of branching Brownian motion with absorption

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly

The number of absorbed individuals in branching Brownian motion with a barrier

We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $x > 0$, we add an absorbing barrier, i.e.\ individuals touching the

Survival probabilities for branching Brownian motion with absorption

We study a branching Brownian motion (BBM) with absorption, in which particles move as Brownian motions with drift $-\rho$, undergo dyadic branching at rate $\beta>0$, and are killed on hitting the

A note on speed of convergence to the quasi-stationary distribution

Abstract In this note we show that for Z being a birth and death process on ℤ or Brownian motion with drift and ℰ′ = (0, ∞), the speed of convergence to the quasistationary distribution is of order

On the Kesten-Stigum theorem in $L^2$ beyond $\lambda$-positivity

We study supercritical branching processes in which all particles evolve according to some general Markovian motion (which may possess absorbing states) and branch independently at a fixed constant

Branching Brownian motion seen from its tip

It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052–1061, 1987) that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an

Branching brownian motion with absorption