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On two variant models of branching Brownian motion


Branching Brownian motion is a random particle system which incorporates both the tree-like structure and the diffusion process. We consider two variant models of branching Brownian motion, branching Brownian motion with absorption and branching Brownian motion with an inhomogeneous branching rate.

In the first variant model, branching Brownian motion with absorption, particles move as Brownian motions with drift -\rho, undergo dyadic branching at rate 1, and are killed when they reach the origin. Kesten (1978) first introduced this model and showed that \rho=\sqrt{2} is the critical value separating the supercritical case \rho<\sqrt{2} and the subcritical case \rho>\sqrt{2}. We study the transition of the model from the slightly subcritical regime to the critical regime. Write \rho=\sqrt{2+2\varepsilon}. We obtain a Yaglom type asymptotic result, showing that the long-run expected number of particles conditioned on survival grows exponentially as 1/\sqrt{\varepsilon} as the process gets closer to being critical.

In the second variant model, branching Brownian motion with an inhomogeneous branching rate, each particle independently moves as Brownian motion with negative drift, each particle can die or undergo dyadic fission, and the difference between the birth rate and the death rate is proportional to the particle’s location. This model was first considered by Roberts and Schweinsberg (2021) and models the evolution of populations undergoing selection. Aiming to understand the distribution of fitness levels of individuals in a large population undergoing selection, we study the particle configurations of this model from the left edge to the right edge. We show that, under certain assumptions, after a sufficiently long time, the distribution of individual fitnesses from the least fit individuals to the most fit individuals is approximately a traveling wave with a profile related to the Airy function. Our work, complements the results in Roberts and Schweinsberg (2021), giving a fuller picture of the fitness distribution.

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