Maximal displacement and population growth for branching Brownian motions

Abstract

We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the non-extinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.

Limiting Distribution of Particles Near the Frontier in the Catalytic Branching Brownian Motion

“…Let us mention that versions of (1.2) -(1.4) for a large class of branching Brownian motions were recently proved in [18] and [19]. Also, a while ago, versions of (1.6) and (1.7) for branching Brownian motions with branching rates given by continuous functions decaying sufficiently fast at infinity were proved in [12] and [16] respectively.…”

“…The cases λ = β and λ > β will require separate analysis. Partial results are available in [19] (Theorem 3.7).…”

“…This is made precise by the theory of additive functionals of Brownian motion. See, for example, papers of Chen and Shiozawa [9] and Shiozawa [17], [18], [19] where they study a large class of processes with branching rates which are allowed to be measures.…”

Limiting distributions for particles near the frontier of spatially inhomogeneous branching Brownian motions

“…We assume that both ν + and ν − are compactly supported in R d . By the proof of [24,Theorem 5.2] or [22,Appendix A.1], there exist positive constants c 1 and c 2 such that…”

“…In this subsection, we introduce the branching Brownian motion (see [12,13,14] and [21,22] for details).…”