Spread Rate of Branching Brownian Motions

Abstract

We find the exponential growth rate of the population outside a ball with time dependent radius for a branching Brownian motion in Euclidean space. We then see that the upper bound of the particle range is determined by the principal eigenvalue of the Schrödinger type operator associated with the branching rate measure and branching mechanism. We assume that the branching rate measure is small enough at infinity, and can be singular with respect to the Lebesgue measure. We finally apply our results to several concrete models.

Limiting distribution of the rightmost particle in catalytic branching Brownian motion

“…Let us mention that versions of (1.1) and (1.2) for BBM with branching rate given by a continuous function decaying sufficiently fast at infinity were proved in [7] and [9] respectively. Versions of (1.1) and (1.2) for BBM with branching rate given by measures decaying sufficiently fast at infinity were proved in [14] and [10] respectively. Versions of (1.1) and (1.2) for discretetime catalytic branching random walks on Z were proved in [6].…”

Maximal displacement and population growth for branching Brownian motions

“…We are concerned with the population growth rate related to the maximal displacement for a spatially inhomogeneous branching Brownian motion in Euclidean space R d . We proved in [35] that under the non-extinction condition, this rate is given in terms of the principal eigenvalue of an associated Schrödinger type operator. This result implies the existence of the phase transition for the growth rate.…”

“…Our refinement on the upper deviation type probability of the maximal displacement (Theorem 3.7) is regarded as a spatially inhomogeneous counterpart of Chauvin and Rouault [11]. In particular, we determine the exponential decay rate of this probability more precisely than [35], and bound the polynomial order. Our argument is also similar to that of [11].…”

“…In particular, we see that for d ≥ 3, such population grows polynomially with dimension dependent growth rate. Our approach for Theorem 3.9 is a refinement of that applied to the non-critical case in [35]. To derive the polynomial growth, we make use of the long time behavior of the Feynman-Kac functional associated with a positive measure (see (4.2) below).…”

“…Acknowledgments The author would like to thank Professor Naotaka Kajino, Professor Tomoyuki Shirai, Professor Ryokichi Tanaka, and one of the referees of [35] for their valuable comments motivating this work. He is grateful to Professor Xia Chen for his comment which improved Theorem 3.9.…”