Abstract
It is well-known that the maximal particle in a branching Brownian motion sits near $\sqrt2 t - \frac{3}{2\sqrt2}\log t$ at time $t$. One may then ask about the paths of particles near the frontier: how close can they stay to this critical curve? Two different approaches to this question have been developed. We improve upon the best-known bounds in each case, revealing new qualitative features including marked differences between the two approaches.
Highlights
A standard branching Brownian motion (BBM) begins with one particle at the origin. This particle moves as a Brownian motion, until an independent exponentially distributed time of parameter 1, at which point it is instantaneously replaced by two new particles
One of the most striking results on BBM was given by Bramson [3], who calculated fine asymptotics for the distribution of M (t), providing new results on travelling wave solutions to the FKPP equation; Hu and Shi [10] more recently showed fluctuations in the almost sure behaviour of M (t) on the log scale
As mentioned above, Jaffuel considered branching random walks, but it is not difficult either to adapt his proof, or to apply his result together with standard tightness properties of Brownian motion, to achieve the same upper bound for BBM
Summary
A standard branching Brownian motion (BBM) begins with one particle at the origin This particle moves as a Brownian motion, until an independent exponentially distributed time of parameter 1, at which point it is instantaneously replaced by two new particles. Our first result is that is ν(fAc ) > 0 (which was previously unknown), but particles may stay far above the curve fAc. Secondly, we are able to give much finer asymptotics for Λ(t), both in distribution and almost surely. As mentioned above, Jaffuel considered branching random walks, but it is not difficult either to adapt his proof, or to apply his result together with standard tightness properties of Brownian motion, to achieve the same upper bound for BBM. In order to apply our methods we must only insist that the distribution of this random number has a finite second moment
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