Abstract
In this work we study a nonlocal version of the FisherKPP equation, (Formula presented.) and its relation to a branching Brownian motion with decay of mass as introduced in AddarioBerry 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 FisherKPP equation. We then harness this to obtain several new results concerning the behavior of the particle system.
Original language  English 

Pages (fromto)  24872577 
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  15 Oct 2019 
ASJC Scopus subject areas
 General Mathematics
 Applied Mathematics
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Sarah Penington
 Department of Mathematical Sciences  Royal Society Research Fellow (and Proleptic Reader)
 Probability Laboratory at Bath
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching, Researcher