TY - JOUR

T1 - Fisher-KPP equation with small data and the extremal process of branching Brownian motion

AU - Mytnik, Leonid

AU - Roquejoffre, Jean Michel

AU - Ryzhik, Lenya

N1 - Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2022/2/12

Y1 - 2022/2/12

N2 - We consider the limiting extremal process X of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale Z, the rescaled “density” of particles, which are at distance n+x from a position close to the tip of X, converges in probability to a multiple of the exponential ex as n→+∞. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a 1-stable random variable. Our approach uses analytic techniques and is motivated by the connection between the properties of the branching Brownian motion and the Bramson shift of the solutions to the Fisher-KPP equation with some specific initial conditions initiated in [9,10] and further developed in the present paper. The proofs of the limit theorems for X rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of “size” 0<ε≪1, up to terms of the order [(logε−1)]−1−γ, with some γ>0.

AB - We consider the limiting extremal process X of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale Z, the rescaled “density” of particles, which are at distance n+x from a position close to the tip of X, converges in probability to a multiple of the exponential ex as n→+∞. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a 1-stable random variable. Our approach uses analytic techniques and is motivated by the connection between the properties of the branching Brownian motion and the Bramson shift of the solutions to the Fisher-KPP equation with some specific initial conditions initiated in [9,10] and further developed in the present paper. The proofs of the limit theorems for X rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of “size” 0<ε≪1, up to terms of the order [(logε−1)]−1−γ, with some γ>0.

KW - Branching Brownian motion

KW - Extremal process

KW - Fisher-KPP equation

UR - http://www.scopus.com/inward/record.url?scp=85119674475&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2021.108106

DO - 10.1016/j.aim.2021.108106

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AN - SCOPUS:85119674475

SN - 0001-8708

VL - 396

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108106

ER -