Almost sure path properties of Branching Diffusion Processes

Abstract

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Branching Brownian Motion: Almost Sure Growth Along Scaled Paths

“…We note that the proof by Git [2] works up to this point; the rest of the proof of the upper bound will be concerned with plugging the gap in [2].…”

“…For any closed set D ⊆ C[0, 1] and θ ∈ [0, 1], we have Appendix: The oversight in [2] In [2] it is written that under a certain assumption, setting…”

“…We then apply these tools to give an entirely new proof of the lower bound for Theorem 1 in Section 3. Finally, in Section 4, we take the same approach as in [2] to gain the upper bound along lattice times, and then rule out some technicalities in order to move to continuous time.…”

The unscaled paths of branching Brownian motion

“…This implies that for any δ > 0, almost surely lim sup By the definition of ε above, for any u ∈N f,L (t) the cosine term in ζ f,γL u (t) is at least ε (since the particle is within L of f (t) at time t). Applying inequality (2) we see that…”

“…The classical scaled path properties of branching Brownian motion (BBM) have now been well-studied: for example, see Lee [13] and Hardy and Harris [3] for large deviation results on "difficult" paths which have a small probability of any particle following them, and Git [2] and Harris and Roberts [6] for the almost sure growth rate of the number of particles near "easy" paths along which we see exponential growth in the number of particles. To give these results, the paths of a BBM are rescaled onto the interval [0, 1], echoing the approach of Schilder's theorem for a single Brownian motion.…”