# Conditional speed of branching Brownian motion, skeleton decomposition and application to random obstacles

@article{Oz2015ConditionalSO,
title={Conditional speed of branching Brownian motion, skeleton decomposition and application to random obstacles},
author={Mehmet Oz and Mine cCauglar and J'anos Englander},
journal={arXiv: Probability},
year={2015}
}
• Published 5 July 2015
• Mathematics
• arXiv: Probability
We study a branching Brownian motion $Z$ in $\mathbb{R}^d$, among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the whole motion when hit by a particle. Considering a general offspring distribution, we derive the decay rate of the annealed probability that none of the particles of $Z$ hits a trap, asymptotically in time $t$. This proves to be a rich problem motivating the…
6 Citations

## Figures from this paper

### Branching Brownian motion in an expanding ball and application to the mild obstacle problem

We first consider a d-dimensional branching Brownian motion (BBM) evolving in an expanding ball, where the particles are killed at the boundary of the ball, and the expansion is subdiffusive in time.

### Large deviations for local mass of branching Brownian motion

We study the local mass of a dyadic branching Brownian motion Z evolving in Rd. By ‘local mass,’ we refer to the number of particles of Z that fall inside a ball with fixed radius and time-dependent

### On the density of branching Brownian motion in subcritical balls

We study the density of the support of a dyadic d-dimensional branching Brownian motion (BBM) in subcritical balls in Rd. Using elementary geometric arguments and an extension of a previous result on

### On the volume of the shrinking branching Brownian sausage

• Mehmet Oz
• Mathematics
Electronic Communications in Probability
• 2020
The branching Brownian sausage in R was defined in [4] similarly to the classical Wiener sausage, as the random subset of R scooped out by moving balls of fixed radius with centers following the

### 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

### Maximum of Branching Brownian Motion among mild obstacles

• Mathematics
• 2022
. We study the height of the maximal particle at time t of a one dimensional branching Brownian motion with a space-dependent branching rate. The branching rate is set to zero in ﬁnitely many

## References

SHOWING 1-10 OF 30 REFERENCES

### Asymptotic radial speed of the support of supercritical branching Brownian motion and super-Brownian motion in R^d

It has long been known that the left-most or right-most particle in a one dimensional dyadic branching Brownian motion with constant branching rate β > 0 has almost sure asymptotic speed √ 2β, (cf.

### Survival asymptotics for branching Brownian motion in a Poissonian trap field

• Mathematics
• 2001
In this paper we study a branching Brownian motion on Rd with branching rate in a Poissonian trap eld whose Borel intensity mea sure is such that d dx decays radially with the distance to the origin

### Quenched law of large numbers for branching Brownian motion in a random medium

. We study a spatial branching model, where the underlying motion is d -dimensional ( d ≥ 1) Brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets

### Escape Probabilities for Branching Brownian Motion Among Soft Obstacles

• Mathematics
• 2010
We derive asymptotics for the quenched probability that a critical branching Brownian motion killed at a small rate ε in Poissonian obstacles exits from a large domain. Results are formulated in

### Maximal displacement of branching brownian motion

It is shown that the position of any fixed percentile of the maximal displacement of standard branching Brownian motion in one dimension is 21/2t–3 · 2−3/2 log t + O(1) at time t, the second-order

### Branching diffusions, superdiffusions and random media

Spatial branching processes became increasingly popular in the past decades, not only because of their obvious connection to biology, but also because superprocesses are intimately related to

### An Application of the Backbone Decomposition to Supercritical Super-Brownian Motion with a Barrier

• Mathematics
Journal of Applied Probability
• 2012
The behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki et al. (2011) is analyzed and conclusions regarding the growth in the right most point in the support, analytical properties of the associated one-sided FKPP equation are obtained.

### KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees

• Mathematics
• 1988
SummaryIf Rt is the position of the rightmost particle at time t in a one dimensional branching brownian motion, u(t, x)=P(Rt>x) is a solution of KPP equation: \frac{{\partial u}}{{\partial t}} =

### Spatial Branching in Random Environments and with Interaction

Preliminaries The Spine Construction and the Strong Law of Large Numbers Examples of the Strong Law The Strong Law for a Type of Self-Interaction The Center of Mass Branching Brownian Motion in a