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We study the maximal degree of (sub)critical L{\'e}vy trees which arise as the scaling limits of Bienaym{\'e}-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree along those nodes. Furthermore, we make sense of the distribution of the L{\'e}vy tree conditioned to have a fixed maximal degree. In the case where the L{\'e}vy measure is diffuse, we show that the maximal degree is realized by a unique node whose height is exponentially distributed and we also prove that the conditioned L{\'e}vy tree can be obtained by grafting a L{\'e}vy forest on an independent size-biased L{\'e}vy tree with a degree constraint at a uniformly chosen leaf. Finally, we show that the L{\'e}vy tree conditioned on having large maximal degree converges locally to an immortal tree (which is the continuous analogue of the Kesten tree) in the critical case and to a condensation tree in the subcritical case. Our results are formulated in terms of the exploration process which allows to drop the Grey condition.

We consider a Feller diffusion (Zs, s $\ge$ 0) (with diffusion coefficient $\sqrt$ 2$\beta$ and drift $\theta$ $\in$ R) that we condition on {Zt = at}, where at is a deterministic function, and we study the limit in distribution of the conditioned process and of its genealogical tree as t $\rightarrow$ +$\infty$. When at does not increase too rapidly, we recover the standard size-biased process (and the associated genealogical tree given by the Kesten's tree). When at behaves as $\alpha$$\beta$ 2 t 2 when $\theta$ = 0 or as $\alpha$ e 2$\beta$|$\theta$|t when $\theta$ = 0, we obtain a new process whose distribution is described by a Girsanov transformation and equivalently by a SDE with a Poissonian immigration. Its associated genealogical tree is described by an infinite discrete skeleton (which does not satisfy the branching property) decorated with Brownian continuum random trees given by a Poisson point measure. As a by-product of this study, we introduce several sets of trees endowed with a Gromovtype distance which are of independent interest and which allow here to define in a formal and measurable way the decoration of a backbone with a family of continuum random trees.

We give an invariance principle for very general additive functionals of conditioned Bienaym{��}-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable L{��}vy tree. This includes the case when the offspring distribution has finite variance (the L{��}vy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height.

International audience; We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton-Watson processes with a penalizing function of the form $P (x)s^x$ where P is a polynomial of degree p and s ∈ [0, 1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s = 1 (or s → 1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton-Watson tree with p distinguished infinite spines.

Under minimal condition, we prove the local convergence of a critical multi-type Galton-Watson tree conditioned on having a large total progeny by types towards a multi-type Kesten's tree. We obtain the result by generalizing Neveu's strong ratio limit theorem for aperiodic random walks on Z^d .

The main object of this course given in Hammamet (December 2014) is the so-called Galton-Watson process.We introduce in the first chapter of this course the framework of discrete random trees. We then use this framework to construct GW trees that describe the genealogy of a GW process. It is very easy to recover the GW process from theGW tree as it is just the number of individuals at each generation. We then give alternativeproofs of classical results on GW processes using the tree formalism. We focus in particular onthe extinction probability (which was the first question of F. Galton) and on the description ofthe processes conditioned on extinction or non extinction.In a second chapter, we focus on local limits of conditioned GW trees. In the critical andsub-critical cases, the population becomesa.s. extinct and the associated genealogical tree is finite. However, it has a small but positiveprobability of being large (this notion must be precised). The question that arises is to describethe law of the tree conditioned of being large, and to say what exceptional event has occurredso that the tree is not typical. A first answer to this question is due to H. Kesten whoconditioned a GW tree to reach height n and look at the limit in distribution when n tends toinfinity. There are however other ways of conditioning a tree to be large: conditioning on havingmany vertices, or many leaves... We present here very recent general results concerning this kindof problems due to the authors of this course and completed by results of X. He., Comment: Lecture given in Hammamet, December 2014

International audience; We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton-Watson tree conditioned on having a large number of protected nodes.

International audience; We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a locally finite measure. We prove that this space with the extended Gromov-Hausdorff-Prokhorov metric is a Polish space. This generalization is needed to define Lévy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.

International audience; By considering a continuous pruning procedure on Aldous's Brownian tree, we construct a random variable $\Theta$ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in a critical Galton-Watson tree. We also prove that this random variable can be obtained as the a.s. limit of the number of cuts needed to cut down the subtree of the continuum tree spanned by $n$ leaves.

Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the $��(3/2,1/2)$-coalescent. By considering a pruning procedure on stable Galton-Watson tree with $n$ labeled leaves, we give a representation of the discrete $��(1+��,1-��)$-coalescent, with $��\in [1/2,1)$ starting from the trivial partition of the $n$ first integers. The construction can also be made directly on the stable continuum L{��}vy tree, with parameter $1/��$, simultaneously for all $n$. This representation allows to use results on the asymptotic number of coalescence events to get the asymptotic number of cuts in stable Galton-Watson tree (with infinite variance for the reproduction law) needed to isolate the root. Using convergence of the stable Galton-Watson tree conditioned to have infinitely many leaves, one can get the asymptotic distribution of blocks in the last coalescence event in the $��(1+��,1-��)$-coalescent.

Considering a random binary tree with $n$ labelled leaves, we use a pruning procedure on this tree in order to construct a $��(3/2,1/2)$-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the $��$-coalescent process up to some time change. These two constructions unable us to obtain results on the coalescent process such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.

We perform a pruning procedure on a L��vy tree and instead of throwing away the removed sub-tree, we regraft it on a given branch (not related to the L��vy tree). We prove that the tree constructed by regrafting is distributed as the original L��vy tree, generalizing a result where only Aldous's tree is considered. As a consequence, we obtain that the quantity which represents in some sense the number of cuts needed to isolate the root of the tree, is distributed as the height of a leaf picked at random in the L��vy tree.

We give an explicit construction of the increasing tree-valued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on L��vy trees. We use the Poissonian structure of the jumps of the increasing tree-valued process to describe its behavior at the first time the tree grows higher than a given height. We also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree-valued process.

Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated L��vy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using L��vy snake techniques. We then prove that the resulting sub-tree after pruning is still a L��vy continuum random tree. This last result is proved using the exploration process that codes the CRT, a special Markov property and martingale problems for exploration processes. We finally give the joint law under the excursion measure of the lengths of the excursions of the initial exploration process and the pruned one.

International audience; In this paper, we propose an edge detection technique based on some local smoothing of the image followed by a statistical hypothesis testing on the gradient. An edge point being defined as a zero-crossing of the Laplacian, it is said to be a significant edge point if the gradient at this point is larger than a threshold $s(\eps)$ defined by: if the image $I$ is pure noise, then $\P(\norm{\nabla I}\geq s(\eps) \bigm| \Delta I = 0) \leq\eps$. In other words, a significant edge is an edge which has a very low probability to be there because of noise. We will show that the threshold $s(\eps)$ can be explicitly computed in the case of a stationary Gaussian noise. In images we are interested in, which are obtained by tomographic reconstruction from a radiograph, this method fails since the Gaussian noise is not stationary anymore. But in this case again, we will be able to give the law of the gradient conditionally on the zero-crossing of the Laplacian, and thus compute the threshold $s(\eps)$. We will end this paper with some experiments and compare the results with the ones obtained with some other methods of edge detection.

We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity proportional to the size of the population. First we establish a Williams' decomposition of the genealogy of the total population given by a continuous random tree, according to the ancestral lineage of the last individual alive. This allows us give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population.

We consider the height process of a Levy process with no negative jumps, and its associated continuous tree representation. Using Levy snake tools developed by Duquesne and Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is interesting in itself.

We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton-Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general sub-critical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates., Comment: 31 pages, 1 figure

We consider the Brownian tree introduced by Aldous and the associated Q-process which consists in an infinite spine on which are grafted independent Brownian trees. We present a reversal procedure on these trees that consists in looking at the tree downward from its top: the branching points becoming leaves and leaves becoming branching points. We prove that the distribution of the tree is invariant under this reversal procedure, which provides a better understanding of previous results from Bi and Delmas (2016)., Comment: This is the second version of the preprint arXiv:1612.03715 that has been splitted into two papers