Excursion theory for Brownian motion indexed by the Brownian tree

Abstract

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...

Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding

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

“…are functions of the jumps of U ′ and U ′′ , respectively, over the time interval [0, T 0 ]. Hence, we can use (23) to get, for every z > 0,…”

“…However, for every ε > 0, the event where ℓ 0 ≥ ε has finite N 0 -measure (the distribution of ℓ 0 under N 0 has a density proportional to ℓ −5/3 , cf. [23,Corollary 3.1]) and the statement of the theorem can be formulated as well under the probability measure N 0 (• | ℓ 0 ≥ ε).…”

Brownian geometry

“…In this section, which is based on [49], we consider the random metric space (Θ, ∆) defined in Theorem 15, which is a free Brownian disk with perimeter z under the probability measure N * ,z 0 . For every x ∈ Θ, define the height of x by H(x) = ∆(x, ∂D).…”

“…A similar result holds for the free Brownian disk D: If r > 0 and H(x) denotes the distance from a point x ∈ D to the boundary, connected components of the set {x ∈ D : H(x) > r} are independent free Brownian disks conditionally on their boundary sizes. Finally, in Section 9, we present the very recent results of [49] studying the sequence of boundary sizes of the connected components of {x ∈ D : H(x) > r} as a process parameterized by r. We show that this process is a growthfragmentation process whose distribution is completely determined. The latter result is very closely related to the recent papers [13,12] investigating scaling limits for a similar process associated with triangulations with a boundary.…”