## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 70, 12 pp.

### Limiting distribution of the rightmost particle in catalytic branching Brownian motion

Sergey Bocharov and Simon C. Harris

#### Abstract

We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $\beta \delta _0(\cdot )$, where $\delta _0(\cdot )$ is the Dirac delta function and $\beta $ is some positive constant. We show that the distribution of the rightmost particle centred about $\frac{\beta } {2}t$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [10] for the degenerate case of catalytic branching.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 70, 12 pp.

**Dates**

Received: 12 May 2016

Accepted: 7 September 2016

First available in Project Euclid: 4 October 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1475601101

**Digital Object Identifier**

doi:10.1214/16-ECP22

**Mathematical Reviews number (MathSciNet)**

MR3564217

**Zentralblatt MATH identifier**

1346.60128

**Subjects**

Primary: 60J55: Local time and additive functionals 60J65: Brownian motion [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Brownian motion local time catalytic branching

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Bocharov, Sergey; Harris, Simon C. Limiting distribution of the rightmost particle in catalytic branching Brownian motion. Electron. Commun. Probab. 21 (2016), paper no. 70, 12 pp. doi:10.1214/16-ECP22. https://projecteuclid.org/euclid.ecp/1475601101