## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 6A (2017), 3752-3794.

### The front location in branching Brownian motion with decay of mass

Louigi Addario-Berry and Sarah Penington

#### Abstract

We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles’ masses decay. In standard BBM, we may define the *front displacement* at time $t$ as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from $\Theta(1)$ to $o(1)$. We show that one can find arbitrarily large times $t$ for which this occurs at a distance $\Theta(t^{1/3})$ behind the front displacement for standard BBM.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 6A (2017), 3752-3794.

**Dates**

Received: December 2015

Revised: August 2016

First available in Project Euclid: 27 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1511773663

**Digital Object Identifier**

doi:10.1214/16-AOP1148

**Mathematical Reviews number (MathSciNet)**

MR3729614

**Zentralblatt MATH identifier**

06838106

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Secondary: 60J65: Brownian motion [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 82C22: Interacting particle systems [See also 60K35]

**Keywords**

Branching Brownian motion branching interacting particle systems front propagation consistent maximal displacement Brownian motion

#### Citation

Addario-Berry, Louigi; Penington, Sarah. The front location in branching Brownian motion with decay of mass. Ann. Probab. 45 (2017), no. 6A, 3752--3794. doi:10.1214/16-AOP1148. https://projecteuclid.org/euclid.aop/1511773663