A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate

Matthew I. Roberts, Jason Schweinsberg

Research output: Contribution to journalArticlepeer-review


Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.
Original languageEnglish
Pages (from-to)1-76
Number of pages76
JournalElectronic Journal of Probability
Early online date2021
Publication statusE-pub ahead of print - 2021

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