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Probabilidade e Combinatória

The front location for branching Brownian motion with decay of mass
Louigi Addario-Berry

McGill University
Segunda-feira, 4 de junho de 2018, 10:30
Sala 228

Consider a standard branching Brownian motion whose particles have varying mass. At time t, if a total mass m of particles have distance less than one from a fixed particle x, then the mass of particle x decays at rate m. The total mass increases via branching events: on branching, a particle of mass m creates two identical mass-m particles.

One may define the front of this system as the point beyond which there is a total mass less than one (or beyond which the expected mass is less than one). This model possesses much less independence than standard BBM, and martingales are hard to come by. Using careful tracking of particle trajectories, we prove a hydrodynamic limit theorem showing that at large times the system is well-approximated by a  nonlocal Fisher-KPP integrodifferential equation. This yields an almost sure law of large numbers for the front speed. We also show that, almost surely, there are arbitrarily large times at which the front lags distance ~ c t^{1/3} behind the typical BBM front. At a high level, our argument for the latter may be described as a proof by contradiction combined with fine estimates on the probability Brownian motion stays in a narrow tube of varying width.

This is joint work with Sarah Penington and Julien Berestycki.