Growth-fragmentation processes in Brownian motion indexed by the Brownian tree

Gall, Jean‐François Le; Riera, Armand

doi:10.48550/arxiv.1811.02825

Cited by 2 publications

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“…In particular N 0 (X r > 0) = N 0 (Z r > 0) < ∞ by (9). The discussion in [25,Section 2.4] now shows that the process (X r ) r>0 has a càdlàg modification under N 0 , which we consider from now on. Furthermore the distribution of this càdlàg modification under N 0 can be interpreted as the excursion measure of the continuous-state branching process with branching mechanism φ(u) = 8/3 u 3/2 (the φ-CSBP in short, see [20, Chapter II] for a brief presentation of continuous-state branching processes).…”

Section: The Local Time Atmentioning

confidence: 80%

“…Set ψ(z) = 3/(3 + z) so that ψ is analytic on a neighborhood of 0 in C. Since ψ(0) = 0, we can find an analytic function G defined on a neighborhood of 0 such that zψ(G(z)) = G(z) for |z| small enough. By (25), we must have F (z) = G(z) for Re(z) > 0 and |z| small, and this means that F can be extended to an analytic function on a neighborhood of 0. By the Lagrange inversion theorem, we have then, for every integer n ≥ 1,…”

Section: Conditional Distributions Of the Local Time Atmentioning

confidence: 99%

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Excursion theory for Brownian motion indexed by the Brownian tree

Abraham

Gall

2018

J. Eur. Math. Soc.

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We derive several explicit distributions of functionals of Brownian motion indexed by the Brownian tree. In particular, we give a direct proof of a result of Bousquet-Mélou and Janson identifying the distribution of the density at 0 of the integrated super-Brownian excursion. IntroductionThe main purpose of the present work is to derive certain explicit distributions for the random process which we call Brownian motion indexed by the Brownian tree, which has appeared in a variety of different contexts. As a key tool for the derivation of our main results we use the excursion theory developed in [1] for Brownian motion indexed by the Brownian tree. In many respects, this excursion theory is similar to the classical Itô theory, which applies in particular to linear Brownian motion and has proved a powerful tool for the calculation of exact distributions of Brownian functionals.Let us briefly describe the objects of interest in this work. We define the Brownian tree T ζ as the random compact R-tree coded by a Brownian excursion ζ = (ζ s ) s≥0 distributed according to the (infinite) Itô measure of positive excursions of linear Brownian motion. If σ stands for the duration of the excursion ζ, this coding means that T ζ is the quotient space of [0, σ] for the equivalence relation defined by s ∼ s ′ if and only if ζ s = ζ s ′ = m ζ (s, s ′ ), where m ζ (s, s ′ ) := min{ζ r : s ∧ s ′ ≤ r ≤ s ∨ s ′ }, and this quotient space is equipped with the metric induced by d ζ (s, s ′The volume measure vol(da) on T ζ is defined as the push forward of Lebesgue measure on [0, σ] under the canonical projection, and the root ρ of T ζ is the equivalence class of 0. We note that under the conditioning by σ = 1 (equivalently the total volume is equal to 1) the tree T ζ is Aldous' Brownian Continuum Random Tree (also called the CRT, see [2,3]), up to an unimportant scaling factor 2.Let us turn to Brownian motion indexed by T ζ . Informally, given T ζ , this is the centered Gaussian process (V a ) a∈T ζ such that V ρ = 0 and Var(V a − V b ) = d ζ (a, b) for every a, b ∈ T ζ . This definition is a bit informal since we are dealing with a random process indexed by a random set. These difficulties can be overcome easily by using the Brownian snake approach. We let (W s ) s≥0 be the Brownian snake (whose spatial motion is linear Brownian motion started at 0) driven by the Brownian excursion (ζ s ) s≥0 . Then, for every s ≥ 0, W s is a finite path started at 0 and with lifetime ζ s , and for every a ∈ T ζ we may define V a as the terminal point W s of the path W s , for any s ∈ [0, σ] such that a is the equivalence class of s in T ζ . The Brownian snake approach thus reduces the study of a tree-indexed Brownian motion to that a process indexed by the positive half-line, and we systematically use this approach in the next sections.The total occupation measure Θ(dx) of (V a ) a∈T ζ is the push forward of vol(da) under the mapping a → V a , or equivalently the push forward of Lebesgue measure on [0, σ] under s → W s . Under the special conditioning...

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Section: The Local Time Atmentioning

confidence: 80%

Section: Conditional Distributions Of the Local Time Atmentioning

confidence: 99%

Excursion theory for Brownian motion indexed by the Brownian tree

Abraham

Gall

2018

J. Eur. Math. Soc.

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“…In [7], it was shown that the collection of perimeters of the holes observed when slicing Boltzmann triangulations at all heights converges, when properly rescaled, towards a particular self-similar growth-fragmentation. We also mention Theorems 3 and 23 of Le Gall and Riera [27], which show that, when slicing directly the free Brownian disk, the holes' perimeters are described by the same growth-fragmentation as in [6] (see also [32] Section 4). When X starts from a single cell of size 0 that grows indefinitely, the geometrical connection corresponds this time to the holes in a sliced discrete approximation of the Brownian plane.…”

Section: Introductionmentioning

confidence: 93%

“…Note that we get an additional factor 3/8 compare to (25). The cumulant function κ in [27] Section 11.1 is equal to 8/3 • κ 3/2 , which simply corresponds to multiplying the distance in the Brownian map by a constant. This means in particular, considering this κ instead of ours, that η − becomes 8/3 • η − , and the constant c − of Lemma 13 becomes 8/3 • c − .…”

Section: Application To Random Mapsmentioning

confidence: 99%

Intrinsic area near the origin for self-similar growth-fragmentations and related random surfaces

Ged¹

2019

Preprint

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We study the behaviour of a natural measure defined on the leaves of the genealogical tree of some branching processes, namely self-similar growth-fragmentation processes. Each particle, or cell, is attributed a positive mass that evolves in continuous time according to a positive self-similar Markov process and gives birth to children at negative jumps events. We are interested in the asymptotics of the mass of the ball centered at the root, as its radius decreases to 0. We obtain the almost sure behaviour of this mass when the Eve cell starts with a strictly positive size. This differs from the situation where the Eve cell grows indefinitely from size 0. In this case, we show that, when properly rescaled, the mass of the ball converges in distribution towards a non-degenerate random variable. We then derive bounds describing the almost sure behaviour of the rescaled mass. Those results are applied to certain random surfaces, exploiting the connection between growth-fragmentations and random planar maps obtained in [6]. This allows us to extend a result of Le Gall [24] on the volume of a free Brownian disk close to its boundary, to a larger family of stable disks. The upper bound of the mass of a typical ball in the Brownian map is refined, and we obtain a lower bound as well.

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Growth-fragmentation processes in Brownian motion indexed by the Brownian tree

Cited by 2 publications

References 0 publications

Excursion theory for Brownian motion indexed by the Brownian tree

Excursion theory for Brownian motion indexed by the Brownian tree

Intrinsic area near the origin for self-similar growth-fragmentations and related random surfaces

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