Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion

Abstract

Consider a one-dimensional branching Brownian motion, and rescale the coordinate and time so that the rates of branching and diffusion are both equal to $1$. If $X_1(t)$ is the position of the rightmost particle of the branching Brownian motion at time $t$, the empirical velocity $c$ of this rightmost particle is defined as $c=X_1(t)/t$. Using the Fisher-KPP equation, we evaluate the probability distribution $\mathcal{P}(c,t)$ of this empirical velocity $c$ in the long time $t$ limit for $c>2$. It was already known that, for a single seed particle, $\mathcal{P}(c,t)\sim \exp [-(c^2/4-1)t]$ up to a prefactor that can depend on $c$ and $t$. Here we show how to determine this prefactor. The result can be easily generalized to the case of multiple seed particles and to branching random walks associated to other traveling wave equations.

11 pages, 2 figures

References in corpus (4)

• Universal Order and Gap Statistics of Critical Branching Brownian Motion
• Spatial Extent of Branching Brownian Motion
• Branching Brownian Motion Conditioned on Particle Numbers
• Phenomenological picture of fluctuations in branching random walks

Cited by in corpus (12)

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• WKB theory of large deviations in stochastic populations
• Slower deviations of the branching Brownian motion and of branching random walks
• Particle-number distribution in large fluctuations at the tip of branching random walks
• Fluctuations of a swarm of Brownian bees
• Persistent fluctuations of the swarm size of Brownian bees
• Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion
• Velocity fluctuations of stochastic reaction fronts propagating into an unstable state: strongly pushed fronts
• Anomalous spreading in reducible multitype branching Brownian motion
• Cross-overs of Bramson's shift at the transition between pulled and pushed fronts
• Large deviations for the rightmost position in a branching Brownian motion
• Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian Bees