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# Documents  60J80 | enregistrements trouvés : 60

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## On determinants of random matrices Zeitouni, Ofer | CIRM H

Post-edited

Probability and Statistics

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## Branching random walks and Galton-Watson trees Gantert, Nina | CIRM H

Post-edited

Probability and Statistics

We give some results about tree-indexed random walks aka branching random walks. In particular, we investigate the growth of the maximum of such a walk.
Based on joint work with Piotr Dyszewski and Thomas Hofelsauer.

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## A non exchangeable coalescent arising in phylogenetics Lambert, Amaury | CIRM H

Post-edited

A popular line of research in evolutionary biology is to use time-calibrated phylogenies in order to infer the underlying diversification process. This involves the use of stochastic models of ultrametric trees, i.e., trees whose tips lie at the same distance from the root. We recast some well-known models of ultrametric trees (infinite regular trees, exchangeable coalescents, coalescent point processes) in the framework of so-called comb metric spaces and give some applications of coalescent point processes to the phylogeny of bird species.

However, these models of diversification assume that species are exchangeable particles, and this always leads to the same (Yule) tree shape in distribution. Here, we propose a non-exchangeable, individual-based, point mutation model of diversification, where interspecific pairwise competition is only felt from the part of individuals belonging to younger species. As the initial (meta)population size grows to infinity, the properly rescaled dynamics of species lineages converge to a one-parameter family of coalescent trees interpolating between the caterpillar tree and the Kingman coalescent.

Keywords: ultrametric tree, inference, phylogenetic tree, phylogeny, birth-death process, population dynamics, evolution
A popular line of research in evolutionary biology is to use time-calibrated phylogenies in order to infer the underlying diversification process. This involves the use of stochastic models of ultrametric trees, i.e., trees whose tips lie at the same distance from the root. We recast some well-known models of ultrametric trees (infinite regular trees, exchangeable coalescents, coalescent point processes) in the framework of so-called comb metric ...

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## Freezing and decorated Poisson point processes Zeitouni, Ofer | CIRM H

Post-edited

Probability and Statistics

The freezing in the title refers to a property of point processes: let $\left ( X_i \right )_{i\geq 1}$ denote a point process which is locally finite and has finite maximum. For a function f continuous of compact support, define $Z_f=f\left ( X_1 \right )+f\left ( X_2 \right )+....$ We say that freezing occurs if the Laplace transform of $Z_f$ depends on f only through a shift. I will discuss this notion and its equivalence with other properties of the point process. In particular, such freezing occurs for the extremal process in branching random walks and in certain versions of the (discrete) two dimensional GFF.
Joint work with Eliran Subag
The freezing in the title refers to a property of point processes: let $\left ( X_i \right )_{i\geq 1}$ denote a point process which is locally finite and has finite maximum. For a function f continuous of compact support, define $Z_f=f\left ( X_1 \right )+f\left ( X_2 \right )+....$ We say that freezing occurs if the Laplace transform of $Z_f$ depends on f only through a shift. I will discuss this notion and its equivalence with other ...

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## Séminaire Bourbaki. Vol. 2012/2013: exposés 1059-1073 | Société Mathématique de France 2014

Congrès

- x; 520 p.
ISBN 978-2-85629-785-8

Astérisque , 0361

Localisation : Périodique 1er étage

actions commensurantes # algèbre de Steenrod # algèbres de Lie semi-simple # bases canoniques # biparti # caractère # carte # cartes de mots # catégorification # classification # cohomologie étale # cohomologie galoisienne # cohomologie motivique # commutateurs # complexe des courbes # conjecture de Baum-Connes # conjecture de Bloch-Kato # conjecture de Hodge # conjecture d'Ore # conjecture de Thompson # constantes de Siegel-Veech # corps d'Okounkov # courbure # cycles algébriques # déterminant du laplacien diagramme de Young # différentielles holomorphes # dimension d'Iitaka # distance de Wasserstein # dynamique symbolique # échanges d'intervalles # ÉDP d'évolution # ÉDP stochastiques # endoscopie tordue # équations F-KPP # espace de modules de différentielles quadratiques # espaces métriques mesurés # exposants de Lyapunov # extrêmes # flot de la chaleur # flot géodésique de Teichmüller # flots de gradient # fonction de Hilbert # fonctorialité # formes automorphes de carré intégrable # graphe expanseur # groupe hyperbolique # groupes approximativement finis # groupes classiques # groupes élémentairement moyennables # groupes kleiniens # groupes moyennables # groupes pleins-topologiques # groupes quantiques # homéomorphismes minimaux # hyperbolicité au sens de Kobayashi # inégalités de Morse holomorphes # KK-théorie # K-théorie de Milnor # laminations terminales # mouvement brownien branchant # odomètres # partition # polynôme de Kerov propriété (T) # renormalisation # sous-décalages topologiques # surfaces plates # symétriseur de Young # théorie homotopique des schémas # trajectoires rugueuses # unicellulaire # variations de structure de Hodge actions commensurantes # algèbre de Steenrod # algèbres de Lie semi-simple # bases canoniques # biparti # caractère # carte # cartes de mots # catégorification # classification # cohomologie étale # cohomologie galoisienne # cohomologie motivique # commutateurs # complexe des courbes # conjecture de Baum-Connes # conjecture de Bloch-Kato # conjecture de Hodge # conjecture d'Ore # conjecture de Thompson # constantes de Siegel-Veech # corps ...

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## Probability and real trees.Ecole d'été de probabilités de Saint-Flour XXXV-2005 Evans, Steven N. | Springer 2008

Congrès

- xi; 193 p.
ISBN 978-3-540-74797-0

Lecture notes in mathematics , 1920

Localisation : Collection 1er étage

arbre aléatoire # propriété asymptotique # convergence de mesure de probabilité # distance de Gromov-Hausdorff # forme de Dirichlet

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## Workshop on branching processes and their applications.Selected papers based on the presentations at the workshop WBPA 09Badajoz # april 20-23, 2009 Gonzalez, Miguel ; del Puerto, Inés M. ; Martinez, Rodrigo ; Molina, Manuel ; Mota, Manuel ; Ramos, Alfonso | Springer 2010

Congrès

- xix; 296 p.
ISBN 978-3-642-11154-9

Lecture notes in statistics - proceedings , 0197

Localisation : Colloque 1er étage (BADA)

processus de branchement # applications # épidémiologie # génétique # cinétique cellulaire # dynamique des popullations # théorème limite

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## Séminaire de probabilités XLII Donati-Martin, Catherine ; Emery, Michel ; Rouault, Alain ; Stricker, Christophe | Springer 2009

Congrès

- xiii; 449 p.
ISBN 978-3-642-01762-9

Lecture notes in mathematics , 1979

Localisation : Collection 1er étage

Probabilités # Processus stochastiques # Distribution # Théorie des systèmes # théorie des opérateurs # théorie des nombres # équation fonctionnelle

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## II Simposio de probabilidad y procesos estocasticosI Encuentro Mexico-Chile de analysis estocastico Caballero, M. E. ; Gorostiza, L. G. | Sociedad Mathematica Mexicana 1992

Congrès

ISBN 978-968-36-2794-0

Aportaciones matematicas , 0007

Localisation : Salle de manutention

analyse statistique # densité de la solution super-processus # embranchement # large déviation # probabilité statistique # problème de Cauchy # équation de Dirac # équation de la diffusion # équation différentielle stochastique

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## Les cahiers de mathématiques de Paris X Nanterre | U.E.R. de Sciences Economiques Université Paris X Nanterre 1986

Congrès

- pag. mult.

Les cahiers de mathématiques Paris X Nanterre , 0001

Localisation : Salle de manutention

construction de tests d'hypothèses # détection statistique d'effet perturbateur # espace mesurable muni de filtration # marquage d'arbre # observation visuelle d'astrolabe # processus de branchement # singularité de loi # utilisaiton de quantile conditionnel # économie mathématique

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## Uma introdução aos processos estocásticos espaciais24-28 Julho Schinazi, Rinaldo | IMPA 1995

Congrès

ISBN 978-85-244-0091-9

Localisation : Salle de manutention

chaîne de Markov à branchement de temps continu # chaîne de Markov à temps discret # espace de probabilités # indépendance # marche aléatoire # première transition de phase continue # probabilité sur espace dénombrable # processus de branchement de Bienaymé-Galton-Watson # processus de contact sur arbre homogène # processus stochastique spatial # seconde transition de phase discontinue # variable aléatoire discrète

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## Seminar on stochastic processes, 19918th seminar on stochastic processes, held at the University of California March 23-25 Chung, K. L. ; Cinlar, E. ; Sharpe, M. J. | Birkhäuser 1992

Congrès

ISBN 978-0-8176-3628-9

Progress in probability , 0029

Localisation : Colloque 1er étage (LOS)

analyse classique # chaîne de Markov # fonction harmonique # martingale # mouvement brownien # processus aléatoire # processus de Markov # processus de branchement # processus stochastique

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## Branching processespapers presented at a conference held at the University of Montréal Aug. 11-20 Joffe, Anatole ; Ney, Peter d. | Marcel Dekker, Inc. 1978

Congrès

ISBN 978-0-8247-6800-3

Advances in probability and related topics , 0005

Localisation : Colloque 1er étage (SAIN)

approximation # diffusion de branchement multigroupe # distribution spatiale # frontière de Martin # inférence statistique # mesure aléatoire # méthode analytique # méthode de martingale # principe d

60J80

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## Condensation in random trees 3/3 Kortchemski, Igor | CIRM H

Multi angle

Probability and Statistics

We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathematical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.). We shall specifically focus on Bienaymé-Galton-Watson trees (which is the simplest
possible genealogical model, where individuals reproduce in an asexual and stationary way), whose offspring distribution is subcritical and is regularly varying. The main tool is to code these trees by integer-valued random walks with negative drift, conditioned on a late return to the origin. The study of such random walks, which is of independent interest, reveals a "one-big jump principle" (identified by Armendáriz & Loulakis in 2011), thus explaining the condensation phenomenon.

Section 1 gives some history and motivations for studying Bienaymé-Galton-Watson trees.
Section 2 defines Bienaymé-Galton-Watson trees.
Section 3 explains how such trees can be coded by random walks, and introduce several useful tools, such as cyclic shifts and the Vervaat transformation, to study random walks under a conditioning involving positivity constraints.
Section 4 contains exercises to manipulate connections between BGW trees and random walks, and to study ladder times of downward skip-free random walks.
Section 5 gives estimates, such as maximal inequalities, for random walks in order to establish a "one-big jump principle".
Section 6 transfers results on random walks to random trees in order to identity the condensation phenomenon.

The goal of these lecture notes is to be as most self-contained as possible.
We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms ...

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## Condensation in random trees 2/3 Kortchemski, Igor | CIRM H

Multi angle

Probability and Statistics

We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathematical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.). We shall specifically focus on Bienaymé-Galton-Watson trees (which is the simplest
possible genealogical model, where individuals reproduce in an asexual and stationary way), whose offspring distribution is subcritical and is regularly varying. The main tool is to code these trees by integer-valued random walks with negative drift, conditioned on a late return to the origin. The study of such random walks, which is of independent interest, reveals a "one-big jump principle" (identified by Armendáriz & Loulakis in 2011), thus explaining the condensation phenomenon.

Section 1 gives some history and motivations for studying Bienaymé-Galton-Watson trees.
Section 2 defines Bienaymé-Galton-Watson trees.
Section 3 explains how such trees can be coded by random walks, and introduce several useful tools, such as cyclic shifts and the Vervaat transformation, to study random walks under a conditioning involving positivity constraints.
Section 4 contains exercises to manipulate connections between BGW trees and random walks, and to study ladder times of downward skip-free random walks.
Section 5 gives estimates, such as maximal inequalities, for random walks in order to establish a "one-big jump principle".
Section 6 transfers results on random walks to random trees in order to identity the condensation phenomenon.

The goal of these lecture notes is to be as most self-contained as possible.
We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms ...

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## Condensation in random trees 1/3 Kortchemski, Igor | CIRM H

Multi angle

Probability and Statistics

We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms for instance connected with data allocation), combinatorics (trees are combinatorial objects by essence), mathematical genetics (as phylogenetic trees), in statistical physics (for instance in connection with random maps as we will see below) and in probability theory (where trees describe the genealogical structure of branching processes, fragmentation processes, etc.). We shall specifically focus on Bienaymé-Galton-Watson trees (which is the simplest
possible genealogical model, where individuals reproduce in an asexual and stationary way), whose offspring distribution is subcritical and is regularly varying. The main tool is to code these trees by integer-valued random walks with negative drift, conditioned on a late return to the origin. The study of such random walks, which is of independent interest, reveals a "one-big jump principle" (identified by Armendáriz & Loulakis in 2011), thus explaining the condensation phenomenon.

Section 1 gives some history and motivations for studying Bienaymé-Galton-Watson trees.
Section 2 defines Bienaymé-Galton-Watson trees.
Section 3 explains how such trees can be coded by random walks, and introduce several useful tools, such as cyclic shifts and the Vervaat transformation, to study random walks under a conditioning involving positivity constraints.
Section 4 contains exercises to manipulate connections between BGW trees and random walks, and to study ladder times of downward skip-free random walks.
Section 5 gives estimates, such as maximal inequalities, for random walks in order to establish a "one-big jump principle".
Section 6 transfers results on random walks to random trees in order to identity the condensation phenomenon.

The goal of these lecture notes is to be as most self-contained as possible.
We study a particular family of random trees which exhibit a condensation phenomenon (identified by Jonsson & Stefánsson in 2011), meaning that a unique vertex with macroscopic degree emerges. This falls into the more general framework of studying the geometric behavior of large random discrete structures as their size grows. Trees appear in many different areas such as computer science (where trees appear in the analysis of random algorithms ...

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## Branching for PDEs Warin, Xavier | CIRM H

Multi angle

Partial Differential Equations;Probability and Statistics

Branching methods have recently been developed to solve some PDEs. Starting from Mckean formulation, we give the initial branching method to solve the KPP equation. We then give a formulation to solve non linear equation with a non linearity polynomial in the value function u. The methodology is extended for general non linearities in the value function u. Then we develop the methodology to solve non linear equation with non linearities polynomial in u and Du with convergence results. At last we give some numerical schemes to solve the semi-linear case and even the full non linear case but currently without convergence results. Branching methods have recently been developed to solve some PDEs. Starting from Mckean formulation, we give the initial branching method to solve the KPP equation. We then give a formulation to solve non linear equation with a non linearity polynomial in the value function u. The methodology is extended for general non linearities in the value function u. Then we develop the methodology to solve non linear equation with non linearities ...

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## Processus de Pólya à valeur mesure Mailler, Cécile | CIRM H

Multi angle

Probability and Statistics

Une urne de Pólya est un processus stochastique décrivant la composition d'une urne contenant des boules de différentes couleurs. L'ensemble des couleurs est usuellement un ensemble fini {1, ..., d}. A chaque instant n, une boule est tirée uniformément au hasard dans l'urne (notons c sa couleur), remise dans l'urne accompagnée de R(c,i) boules de couleur i pour toute couleur i.
Je présente dans cet exposé une généralisation de ce modèle à un ensemble infini, et même potentiellement indénombrable de couleurs. Dans ce nouveau modèle, la composition de l'urne est une mesure (potentiellement non-atomique) sur un espace Polonais.
Ceci est un travail en collaboration avec Jean-François Marckert.
Une urne de Pólya est un processus stochastique décrivant la composition d'une urne contenant des boules de différentes couleurs. L'ensemble des couleurs est usuellement un ensemble fini {1, ..., d}. A chaque instant n, une boule est tirée uniformément au hasard dans l'urne (notons c sa couleur), remise dans l'urne accompagnée de R(c,i) boules de couleur i pour toute couleur i.
Je présente dans cet exposé une généralisation de ce modèle à un ...

60J80

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## Vanishing corrections for the position of an FKPP front Berestycki, Julien | CIRM H

Multi angle

Partial Differential Equations;Probability and Statistics

The celebrated Fisher-Kolmogorov-Petrovsky-Piscounof equation (FKPP) in one dimension for
$h:\mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}$ is:

$\partial_th = \partial{_x^2}h + h - h^2, h(x, 0) = h_0(x)$.

This equation is a natural description of a reaction-diffusion model (Fisher 1937, Kolmogorov et al. 1937, Aronson 1978). It is also related to branching Brownian motion: for the Heaviside initial condition $h_0 (x) = 1{_x<0}$ , $h(x, t)$ is the probability that the rightmost particle at time t in a branching Brownian motion (BBM) is to the right of $x$.
One of the beauty of this equation is that for initial conditions that decrease sufficiently fast, a front develops, i.e. there exists a centring term $m(t)$ and an asymptotic shape $\omega(x)$ such that

$\lim_{t \to \infty} h(m(t) + x,t) = \omega(x) \in (0, 1).$

Since the original paper of Kolmogorov et al., the position of the front $m(t)$ has been studied intensely, in particular by Bramson. In this talk, I will present some recent results concerning a prediction of Ebert and van Saarloos about the vanishing corrections of this position.
Based on a joint work with E. Brunet.
The celebrated Fisher-Kolmogorov-Petrovsky-Piscounof equation (FKPP) in one dimension for
$h:\mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}$ is:

$\partial_th = \partial{_x^2}h + h - h^2, h(x, 0) = h_0(x)$.

This equation is a natural description of a reaction-diffusion model (Fisher 1937, Kolmogorov et al. 1937, Aronson 1978). It is also related to branching Brownian motion: for the Heaviside initial condition $h_0 (x) = 1{_x<0}$ , $h(x, t)$ is ...

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## Competitive populations with vertical and horizontal transmissions Méléard, Sylvie | CIRM H

Multi angle

Mathematics in Science and Technology;Probability and Statistics

Horizontal transfer of information is recognized as a major process in the evolution and adaptation of population, especially micro-organisms. There is a large literature but the previous models are either based on epidemiological models or population genetics stochastic models with constant population size. We propose a general stochastic eco-evolutionary model of population dynamics with horizontal and vertical transfers, inspired by the transfer of plasmids in bacteria. The transfer rates are either density-dependent (DD) or frequency-dependent (FD) or of Michaelis-Menten form (MM). Our model allows eco-evolutionary feedbacks. In the first part we present a two-traits (alleles or kinds of plasmids, etc.) model with horizontal transfer without mutation and study a large population limit. It’s a ODEs system. We show that the phase diagrams are different in the (DD), (FD) and (MM) cases. We interpret the results for the impact of horizontal transfer on the maintenance of polymorphism and the invasion or elimination of pathogens strains. We also propose a diffusive approximation of adaptation with transfer. In a second part, we study the impact of the horizontal transfer on the evolution. We explain why it can drastically affect the evolutionary outcomes. Joint work with S. Billiard,P. Collet, R. Ferrière, C.V. Tran. Horizontal transfer of information is recognized as a major process in the evolution and adaptation of population, especially micro-organisms. There is a large literature but the previous models are either based on epidemiological models or population genetics stochastic models with constant population size. We propose a general stochastic eco-evolutionary model of population dynamics with horizontal and vertical transfers, inspired by the ...

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