Conditioned Brownian trees

By Jean-François Le Gall, M. Weill. 2005

79 citations recorded by the source corpus.

Abstract We consider a Brownian tree consisting of a collection of one-dimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus yields a convenient representation of the so-called Integrated Super-Brownian Excursion (ISE), which can be viewed as the uniform probability measure on the tree of paths. We discuss different approaches that lead to the definition of the Brownian tree conditioned to stay on the positive half-line. We also establish a Vervaat-like theorem showing that this conditioned Brownian tree can be obtained by re-rooting the unconditioned one at the vertex corresponding to the minimal spatial position. In terms of ISE, this theorem yields the following fact: Conditioning ISE to put no mass on ] − ∞ , − e [ and letting e go to 0 is equivalent to shifting the unconditioned ISE to the right so that the left-most point of its support becomes the origin. We derive a number of explicit estimates and formulas for our conditioned Brownian trees. In particular, the probability that ISE puts no mass on ] − ∞ , − e [ is shown to behave like 2 e 4 / 21 when e goes to 0. Finally, for the conditioned Brownian tree with a fixed height h, we obtain a decomposition involving a spine whose distribution is absolutely continuous with respect to that of a nine-dimensional Bessel process on the time interval [ 0 , h ] , and Poisson processes of subtrees originating from this spine.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3bb5b4e22b8b66ad8fbdd791750392d13941fa12

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Citations in the corpus

An invariance principle for random planar maps (2006, unknown): “…n−1/4dM (r, r′) converges in distribution to Cq(RexU − inf Rex) where U is uniform on [0, 1] and independent of Rex, and this has same distribution as Cq supRex by the properties of symmetry and invariance under uniform re-rooting of the Brownian snake that can be found in Le Gall and Weill (2006).”
The topological structure of scaling limits of large planar maps (2006, background): “These procedures all lead to the same limiting pair (e, Z) which can be described as follows from the original pair (e, Z).”
Random geometry on the sphere (2014, unknown): “One can prove (see in particular [36]) that there exists a unique vertex ρ∗ ∈ Te such that Zρ∗ = min c∈Te Zc.”
Scaling limits of random trees and planar maps (2011, background): “The scaling limits of such pairs are still tractable, and in fact, are simple functionals of (e, Z), as shown in [35, 30].”
Some families of increasing planar maps (2007, methodology): “We borrow some considerations from Le Gall & Weill [31] and Le Gall [28].”
Geodesics in large planar maps and in the Brownian map (2008, background): “See Section 5 of [26] for more details.”
Compact Brownian surfaces I: Brownian disks (2015, background): “g ; i r (t) = p supfs2[a i;b i] : Z s= Z a i tg ; where tranges over [0;Z a i Z a i;b i ] (recall that Z a i = Z b i ). Those are geodesic paths in(D;D) fromp(a i) top(s i ), wheres i isthe(a.s.unique[29])timein [a i;b i] atwhich Zattainsitsinﬁmumonthatsameinterval. Alternatively, thesepathsareparts”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, result): “Thanks to Lemma 25, we will derive Lemma 11 from the following similar result under Nx, which is due to Le Gall and Weill [LGW06].”
A view from infinity of the uniform infinite planar quadrangulation (2012, background): “22] (see also [28] for a similar use of these absolute continuity relations).”
Random trees and applications (2005, background): “There are similar limit theorems for interacting particle systems in Zd: See [5] for the voter model and coalescing random walks and the recent paper [22] for the contact process.”
Uniqueness and universality of the Brownian map (2011, methodology): “ribution of ∆ can be found in [9]. Here we will only use the simple fact that the topological support of the law of ∆ is the whole of R+. This can be veriﬁed by elementary arguments. It is known (see [19], Proposition 2.5) that there is an almost surely unique instant s∗ ∈(0,1) such that Zs∗ =−∆. We”
The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection (2013, background): “nd D˜ 6dZ is a simple consequence of (7) and (10). From this and the deﬁnition of D as the largest pseudo-distance satisfying (9), we obtain that D˜ 6 D. On the other hand, let s∗ be the (a.s. unique [LGW06]) point at which Z attains its minimum. Takingasequence(in)suchthatv˜i n = v∗,itisnotdifﬁcult”
The Brownian map is the scaling limit of uniform random plane quadrangulations (2011, background): “Furthermore, by [19], the process Z attains its overall minimum a.”
On the sphericity of scaling limits of random planar quadrangulations (2007, background): “= For simplicity we let S S The main result of [9] says that the convergence in distribution in C ([0 , 1] , R ) 2 holds: where ( e , Z ) is the Brownian snake conditioned to be positive introduced by Le Gall & Weill [12].”