### Abstract

In this work we study a nonlocal version of the Fisher-KPP equation, (Formula presented.) and its relation to a branching Brownian motion with decay of mass as introduced in Addario-Berry and Penington (2015), i.e., a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in ℝ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighborhood around them (as measured by the function φ). We obtain two types of results. First, we study the behavior of solutions to the partial differential equation above. We show that, under suitable conditions on φ and u
_{0}
, the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik. Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the nonlocal Fisher-KPP equation. We then harness this to obtain several new results concerning the behavior of the particle system.

Original language | English |
---|---|

Pages (from-to) | 2487-2577 |

Number of pages | 91 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 72 |

Issue number | 12 |

Early online date | 19 Apr 2019 |

DOIs | |

Publication status | Published - 1 Dec 2019 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*72*(12), 2487-2577. https://doi.org/10.1002/cpa.21827

**Branching Brownian Motion with Decay of Mass and the Nonlocal Fisher-KPP Equation.** / Addario-Berry, Louigi; Berestycki, Julien; Penington, Sarah.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 72, no. 12, pp. 2487-2577. https://doi.org/10.1002/cpa.21827

}

TY - JOUR

T1 - Branching Brownian Motion with Decay of Mass and the Nonlocal Fisher-KPP Equation

AU - Addario-Berry, Louigi

AU - Berestycki, Julien

AU - Penington, Sarah

PY - 2019/12/1

Y1 - 2019/12/1

N2 - In this work we study a nonlocal version of the Fisher-KPP equation, (Formula presented.) and its relation to a branching Brownian motion with decay of mass as introduced in Addario-Berry and Penington (2015), i.e., a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in ℝ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighborhood around them (as measured by the function φ). We obtain two types of results. First, we study the behavior of solutions to the partial differential equation above. We show that, under suitable conditions on φ and u 0 , the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik. Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the nonlocal Fisher-KPP equation. We then harness this to obtain several new results concerning the behavior of the particle system.

AB - In this work we study a nonlocal version of the Fisher-KPP equation, (Formula presented.) and its relation to a branching Brownian motion with decay of mass as introduced in Addario-Berry and Penington (2015), i.e., a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in ℝ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighborhood around them (as measured by the function φ). We obtain two types of results. First, we study the behavior of solutions to the partial differential equation above. We show that, under suitable conditions on φ and u 0 , the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik. Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the nonlocal Fisher-KPP equation. We then harness this to obtain several new results concerning the behavior of the particle system.

UR - http://www.scopus.com/inward/record.url?scp=85064605397&partnerID=8YFLogxK

U2 - 10.1002/cpa.21827

DO - 10.1002/cpa.21827

M3 - Article

VL - 72

SP - 2487

EP - 2577

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 12

ER -