A simple path to asymptotics for the frontier of a branching Brownian motion

Roberts, Matthew I.

doi:10.1214/12-aop753

Cited by 59 publications

(54 citation statements)

References 18 publications

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“…Related to the so-called Fisher-Kolmogorov-Petrovskii-Piskunov equation, equations like (1.4) and (1.5) are studied more in detail for a class of branching Brownian motions on R (see, e.g., [31,32], [10], [28], [29], [35], [8], [37, Subsection 1.1] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Spread Rate of Branching Brownian Motions

Shiozawa

2017

Acta Appl Math

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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.

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Section: Introductionmentioning

confidence: 99%

Spread Rate of Branching Brownian Motions

Shiozawa

2017

Acta Appl Math

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“…Bramson's original proof was probabilistic. A shorter probabilistic proof can be found in a recent paper [32], while the PDE proofs can be found in [24,36] and, more recently, in [20]. Various extensions to equations with inhomogeneous coefficients have also been studied in [14,15,21,25,28].…”

Section: The Main Resultsmentioning

confidence: 99%

The Bramson logarithmic delay in the cane toads equations

2017

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We study a nonlocal reaction-diffusion-mutation equation modeling the spreading of a cane toads population structured by a phenotypical trait responsible for the spatial diffusion rate. When the trait space is bounded, the cane toads equation admits traveling wave solutions [7]. Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the traveling waves grows as (3/(2λ * )) log t. This result relies on a present-time Harnack inequality which allows to compare solutions of the cane toads equation to those of a Fisher-KPP type equation that is local in the trait variable.with a non-negative compactly supported initial condition v 0 (x) = v(0, x) propagate with the speed c * = 2 in the sense that lim t→+∞ v(t, ct) = 0, (1.6) for all c > c * , and lim t→+∞ v(t, ct) = 1, (1.7)G t = L x G, t > 0, x, y ∈ R n , G(0, ·, y) = δ(· − y), (2.1) so that the solution ofcan be written, for all t > 0 and x ∈ R n , as u(t, x) =ˆR n G(t, x, y)u 0 (y)dy.The notation L x in (2.1) means that the operator L acts on G in the x variable. There are wellknown Gaussian bounds for G (see e.g. [2,13]) of the type c 1 t n/2 e −c 2 |x−y| 2 t ≤ G(t, x, y) ≤ C 1 t n/2 e −C 2 |x−y| 2 t ,

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“…Recently, Roberts simplified the proofs of these results [31]. Ebert and van Saarloos, in [10], obtained, through non-rigorous methods, the next term in the expansion.…”

Section: Such a Constant May Depend On Initial Data Or Various Other mentioning

confidence: 99%

Population stabilization in branching Brownian motion with absorption and drift

Henderson

2016

Communications in Mathematical Sciences

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We consider, through PDE methods, branching Brownian motion with drift and absorption. It is well known that there exists a critical drift which separates those processes which die out almost surely and those which survive with positive probability. In this work, we consider lower order corrections to the critical drift which ensures a non-negative, bounded expected number of particles and convergence of this expectation to a limiting non-negative number, which is positive for some initial data. In particular, we show that the average number of particles stabilizes at the convergence rate O(log(t)/t) if and only if the multiplicative factor of the O(t −1/2 ) correction term is 3 √ πt −1/2 . Otherwise, the convergence rate is O(1/ √ t). We point out some connections between this work and recent work investigating the expansion of the front location for the initial value problem in Fisher-KPP [7,8,15,16].

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A simple path to asymptotics for the frontier of a branching Brownian motion

Abstract: We give short proofs of two classical results about the position of the extremal particle in a branching Brownian motion, one concerning the median position and another the almost sure behaviour.

Cited by 59 publications

References 18 publications

Spread Rate of Branching Brownian Motions

Spread Rate of Branching Brownian Motions

The Bramson logarithmic delay in the cane toads equations

Population stabilization in branching Brownian motion with absorption and drift

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