# random trees, lévy processes and spatial branching processes

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Random Trees, Levy Processesand Spatial Branching Processes

Thomas Duquesne1, Jean-Francois Le Gall2

1CMLA, ENS de Cachan, 61 av. du President Wilson, 94235 CACHAN Cedex,e-mail: Thomas.Duquesne@cmla.ens-cachan.fr

2DMA, ENS, 45, rue dUlm, 75230 PARIS Cedex 05,e-mail: legall@dma.ens.fr

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Contents

0.1 Discrete trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.2 Galton-Watson trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.3 The continuous height process . . . . . . . . . . . . . . . . . . . . . . . 80.4 From discrete to continuous trees . . . . . . . . . . . . . . . . . . . . . 110.5 Duality properties of the exploration process . . . . . . . . . . . . . . 130.6 Marginals of trees coded by the height process . . . . . . . . . . . . . 140.7 The Levy snake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1 The height process 191.1 Preliminaries on Levy processes . . . . . . . . . . . . . . . . . . . . . . 19

1.1.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 191.1.2 Local times at the maximum and the minimum . . . . . . . . . 20

1.2 The height process and the exploration process . . . . . . . . . . . . . 251.3 Local times of the height process . . . . . . . . . . . . . . . . . . . . . 31

1.3.1 The construction of local times . . . . . . . . . . . . . . . . . . 311.3.2 Some properties of local times . . . . . . . . . . . . . . . . . . . 34

1.4 Three applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.4.1 The Ray-Knight theorem . . . . . . . . . . . . . . . . . . . . . 391.4.2 The continuity of the height process . . . . . . . . . . . . . . . 411.4.3 Holder continuity of the height process . . . . . . . . . . . . . . 42

2 Convergence of Galton-Watson trees 452.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.2 The convergence of finite-dimensional marginals . . . . . . . . . . . . . 462.3 The functional convergence . . . . . . . . . . . . . . . . . . . . . . . . 512.4 Convergence of contour processes . . . . . . . . . . . . . . . . . . . . . 582.5 A joint convergence

and an application to conditioned trees . . . . . . . . . . . . . . . . . . 592.6 The convergence of reduced trees . . . . . . . . . . . . . . . . . . . . . 632.7 The law of the limiting reduced tree . . . . . . . . . . . . . . . . . . . 64

3 Marginals of continuous trees 713.1 Duality properties of the exploration process . . . . . . . . . . . . . . 713.2 The tree associated with Poissonnian marks . . . . . . . . . . . . . . . 80

3.2.1 Trees embedded in an excursion . . . . . . . . . . . . . . . . . 803.2.2 Poissonnian marks . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.3 Marginals of stable trees . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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4 The Levy snake 914.1 The construction of the Levy snake . . . . . . . . . . . . . . . . . . . . 91

4.1.1 Construction of the snake with a fixed lifetime process . . . . . 914.1.2 The definition of the Levy snake . . . . . . . . . . . . . . . . . 924.1.3 The strong Markov property . . . . . . . . . . . . . . . . . . . 944.1.4 Excursion measures . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 The connection with superprocesses . . . . . . . . . . . . . . . . . . . 974.2.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . 974.2.2 First step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2.3 Second step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.3 Exit measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4 Continuity properties of the Levy snake . . . . . . . . . . . . . . . . . 1084.5 The Brownian motion case . . . . . . . . . . . . . . . . . . . . . . . . . 1104.6 The law of the Levy snake at a first exit time . . . . . . . . . . . . . . 1154.7 The reduced tree in an open set . . . . . . . . . . . . . . . . . . . . . . 120

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Introduction

The main goal of this work is to investigate the genealogical structure of continuous-state branching processes in connection with limit theorems for discrete Galton-Watson trees. Applications are also given to the construction and various propertiesof spatial branching processes including a general class of superprocesses.

Our starting point is the recent work of Le Gall and Le Jan [32] who proposeda coding of the genealogy of general continuous-state branching processes via a real-valued random process called the height process. Recall that continuous-state branch-ing processes are the continuous analogues of discrete Galton-Watson branching pro-cesses, and that the law of any such process is characterized by a real function called the branching mechanism. Roughly speaking, the height process is a contin-uous analogue of the contour process of a discrete branching tree, which is easy tovisualize (see Section 0.1, and note that the previous informal interpretation of theheight process is made mathematically precise by the results of Chapter 2). In the im-portant special case of the Feller branching diffusion ((u) = u2), the height processis reflected linear Brownian motion: This unexpected connection between branchingprocesses and Brownian motion, or random walk in a discrete setting has been knownfor long and exploited by a number of authors (see e.g. [3], [11], [18], [39], [42]). Thekey contribution of [32] was to observe that for a general subcritical continuous-statebranching process, there is an explicit formula expressing the height process as a func-tional of a spectrally positive Levy process whose Laplace exponent is precisely thebranching mechanism. This suggests that many problems concerning the genealogy ofcontinuous-state branching processes can be restated and solved in terms of spectrallypositive Levy processes, for which a lot of information is available (see e.g. Bertoinsrecent monograph [5]). It is the principal aim of the present work to develop suchapplications.

In the first two sections below, we briefly describe the objects of interest in adiscrete setting. In the next sections, we outline the main contributions of the presentwork.

0.1 Discrete trees

Let

U =n=0

Nn

where N = {1, 2, . . .} and by convention N0 = {}. If u = (u1, . . . , un) Nn, we set|u| = n, so that |u| represents the generation of u. If u = (u1, . . . um) and v =(v1, . . . , vn) belong to U , we write uv = (u1, . . . um, v1, . . . , vn) for the concatenationof u and v. In particular u = u = u.

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A (finite) rooted ordered tree T is a finite subset of U such that:

(i) T .

(ii) If v T and v = uj for some u U and j N, then u T .

(iii) For every u T , there exists a number ku(T ) 0 such that uj T if and onlyif 1 j ku(T ).

We denote by T the set of all rooted ordered trees. In what follows, we see eachvertex of the tree T as an individual of a population whose T is the family tree. Thecardinality #(T ) of T is the total progeny.

If T is a tree and u T , we define the shift of T at u by uT = {v U : uv T }.Note that u T.

We now introduce the (discrete) height function associated with a tree T . Letus denote by u(0) = , u(1), u(2), . . . , u(#(T ) 1) the elements of T listed in lexi-cographical order. The height function H(T ) = (Hn(T ); 0 n < #(T )) is definedby

Hn(T ) = |u(n)|, 0 n < #(T ).

The height function is thus the sequence of the generations of the individuals ofT , when these individuals are visited in the lexicographical order (see Fig.1 for anexample). It is easy to check that H(T ) characterizes the tree T .

AAAAA

AAAAA

AAAAA

1 2

11 12 13

121 122

tree T

DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

contour function height function

CCCCCCCCCC

1 2 3 (T ) 1 2 3 #(T )1

1

2

1

2

Figure 1

The contour function gives another way of characterizing the tree, which iseasier to visualize on a picture (see Fig.1). Suppose that the tree is embedded inthe half-plane in such a way that edges have length one. Informally, we imagine themotion of a particle that starts at time t = 0 from the root of the tree and thenexplores the tree from the left to the right, moving continuously along the edges atunit speed, until it comes back to its starting point. Since it is clear that each edgewill be crossed twice in this evolution, the total time needed to explore the tree is(T ) := 2(#(T )1). The value Ct of the contour function at time t is the distance (onthe tree) between the position of the particle at time t and the root. By conventionCt = 0 if t (T ). Fig.1 explains the definition of the contour function better thana formal definition.

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0.2 Galton-Watson trees

Let be a critical or subcritical offspring distribution. This means that is a prob-ability measure on Z+ such that

k=0

k(k) 1.

We exclude the trivial case where (1) = 1.There is a unique probability distribution Q on T such that

(i) Q(k = j) = (j), j Z+.

(ii) For every j 1 with (j) > 0, the shifted trees 1T , . . . , jT are independent un-der the conditional probability Q( | k = j) and their conditional distributionis Q.

A random tree with distribution Q is called a Galton-Watson tree with offspringdistribution , or in short a -Galton-Watson tree.

Let T1, T2, . . . be a sequence of independent -Galton-Watson trees. We can as-sociate with this sequence a height process obtained by concatenating the heightfunctions of each of the trees T1, T2, . . .. More precisely, for every k 1, we set

Hn = Hn(#(T1)++#(Tk1))(Tk) if #(T1)+ +#(Tk1) n < #(T1)+ +#(Tk).

The process (Hn, n 0) codes the sequence of trees.S