## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 103, 40 pp.

### Branching Brownian motion, mean curvature flow and the motion of hybrid zones

Alison Etheridge, Nic Freeman, and Sarah Penington

#### Abstract

We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $\Lambda $-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as *hybrid zones*. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.

#### Article information

**Source**

Electron. J. Probab., Volume 22 (2017), paper no. 103, 40 pp.

**Dates**

Received: 7 July 2017

Accepted: 17 November 2017

First available in Project Euclid: 7 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1512615692

**Digital Object Identifier**

doi:10.1214/17-EJP127

**Mathematical Reviews number (MathSciNet)**

MR3733661

**Zentralblatt MATH identifier**

06827080

**Subjects**

Primary: 60J85: Applications of branching processes [See also 92Dxx] 92D15: Problems related to evolution

**Keywords**

branching Brownian motion mean curvature flow hybrid zones spatial $\Lambda $-Fleming-Viot population genetics

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Etheridge, Alison; Freeman, Nic; Penington, Sarah. Branching Brownian motion, mean curvature flow and the motion of hybrid zones. Electron. J. Probab. 22 (2017), paper no. 103, 40 pp. doi:10.1214/17-EJP127. https://projecteuclid.org/euclid.ejp/1512615692