We prove several limit theorems that relate coalescent processes to continuous-state branching processes. Some of these theorems are stated in terms of the so-called generalized Fleming-Viot processes, which describe the evolution of a population with fixed size, and are duals to the coalescents with multiple collisions studied by Pitman and others. We first discuss asymptotics when the initial size of the population tends to infinity. In that setting, under appropriate hypotheses, we show that a rescaled version of the generalized Fleming-Viot process converges weakly to a continuous-state branching process. As a corollary, we get a hydrodynamic limit for certain sequences of coalescents with multiple collisions: Under an appropriate scaling, the empirical measure associated with sizes of the blocks converges to a (deterministic) limit which solves a generalized form of Smoluchowski's coagulation equation. We also study the behavior in small time of a fixed coalescent with multiple collisions, under a regular variation assumption on the tail of the measure $ν$ governing the coalescence events. Precisely, we prove that the number of blocks with size less than $εx$ at time $(εν([ε,1]))^{-1}$ behaves like $ε^{-1}λ\_1(]0,x[)$ as $ε\to 0$, where $λ\_1$ is the distribution of the size of one cluster at time 1 in a continuous-state branching process with stable branching mechanism. This generalizes a classical result for the Kingman coalescent.
Sufficient conditions are given for existence and uniqueness in Smoluchowski's coagulation equation, for a wide class of coagulation kernels and initial mass distributions. An example of non-uniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski's equation.
This work exhibits a novel phase transition for the classical stochastic block model (SBM). In addition we study the SBM in the corresponding near-critical regime, and find the scaling limit for the component sizes. The two-parameter stochastic process arising in the scaling limit, an analogue of the standard Aldous' multiplicative coalescent, is interesting in its own right. We name it the (standard) Interacting Multiplicative Coalescent. To the best of our knowledge, this object has not yet appeared in the literature.
The Beta coalescents are stochastic processes modeling the genealogy of a population. They appear as the rescaled limits of the genealogical trees of numerous stochastic population models. In this article, we take interest in the number of blocs at small times in the Beta coalescent. Berestycki, Berestycki and Schweinsberg proved a law of large numbers for this quantity. Recently, Limic and Talarczyk proved that a functional central limit theorem holds as well. We give here a simple proof for an unidimensional version of this result, using a coupling between Beta coalescents and continuous-tim…
Coalescents with multiple collisions (also called Lambda-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with immigration. Imagine an infinite population with immigration labelled at each generation by N:={1,2,...}. Some ancestral lineages cannot be followed backwards after some time because their ancestor is outside the population. The individuals with an immigrant ancestor constitute a distinguished family and we define exchangeable distinguished coalescent processes as a…
The (Ξ, A)-Fleming-Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming-Viot process except that the Kingman's coalescent is replaced by the Ξ-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A. We then investigate its reversibility. We also study both the weak and strong uniqueness of solution to the associated stochastic partial differential equation.
The generalized Fleming-Viot processes were defined in 1999 by Donnelly and Kurtz using a particle model and by Bertoin and Le Gall in 2003 using stochastic flows of bridges. In both methods, the key argument used to characterize these processes is the duality between these processes and exchangeable coalescents. A larger class of coalescent processes, called distinguished coalescents, was set up recently to incorporate an immigration phenomenon in the underlying population. The purpose of this article is to define and characterize a class of probability-measure valued processes called the gen…
A generalized stepping stone model with $Ξ$-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman's coalescent is replaced by $Ξ$-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued $Ξ$-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that i…
Rice University（莱斯大学）；Massachusetts Institute of Technology（麻省理工学院）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
The reconstruction of a species phylogeny from genomic data faces two significant hurdles: 1) the trees describing the evolution of each individual gene--i.e., the gene trees--may differ from the species phylogeny and 2) the molecular sequences corresponding to each gene often provide limited information about the gene trees themselves. In this paper we consider an approach to species tree reconstruction that addresses both these hurdles. Specifically, we propose an algorithm for phylogeny reconstruction under the multispecies coalescent model with a standard model of site substitution. The mu…
Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes, that are fundamental for our results. We relate these new objects with the other previously mentioned families introducing some new bijections. We prove joint Benjamini--Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new fractal random…
Coalescents with multiple collisions, also known as $Λ$-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure $Λ$ is the $\operatorname {Beta}(2-α,α)$ distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here, we use a recent result of Birkner et al. to prove that Beta-coalescents can be embedd…
There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Peña \cite{d} establishes a nice exponential inequality for discrete time locally square integrable martingale . In this paper, we obtain de la Peña's inequalities for stochastic integral of multivariate point processes. The proof is primarily based on Doléans-Dade exponential formula and the optional stopping theorem. As application, we obtain an exponential inequality for block counting process in $Λ-$coalescents.
Branching processes and Fleming-Viot processes are two main models in stochastic population theory. Incorporating an immigration in both models, we generalize the results of Shiga (1990) and Birkner et al. (2005) which respectively connect the Feller diffusion with the classical Fleming-Viot process and the alpha-stable continuous state branching process with the Beta(2-alpha, alpha)-generalized Fleming-Viot process. In a recent work, a new class of probability-measure valued processes, called M-generalized Fleming-Viot processes with immigration, has been set up in duality with the so-called …
We approximate stochastic processes in finite dimension by dynamical systems. We provide trajectorial estimates which are uniform with respect to the initial condition for a well chosen distance. This relies on some non-expansivity property of the flow, which allows to deal with non-Lipschitz vector fields. We use the stochastic calculus and follow the martingale technics initiated in Berestycki and al [5] to control the fluctuations. Our main applications deal with the short time behavior of stochastic processes starting from large initial values. We state general properties on the coming dow…
Genetic data are often used to infer demographic history and changes or detect genes under selection. Inferential methods are commonly based on models making various strong assumptions: demography and population structures are supposed \textit{a priori} known, the evolution of the genetic composition of a population does not affect demography nor population structure, and there is no selection nor interaction between and within genetic strains. In this paper, we present a stochastic birth-death model with competitive interactions and asexual reproduction. We develop an inferential procedure fo…
We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions, that is, generalized Fleming--Viot flows and flows of continuous-state branching processes with immigration. One of them unifies the different treatments of three kinds of flows in Bertoin and Le Gall [Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 3…
This paper is the third part of our study started with Cattiaux, León and Prieur [Stochastic Process. Appl. 124 (2014) 1236-1260; ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359-384]. For some ergodic Hamiltonian systems, we obtained a central limit theorem for a nonparametric estimator of the invariant density [Stochastic Process. Appl. 124 (2014) 1236-1260] and of the drift term [ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 359-384], under partial observation (only the positions are observed). Here, we obtain similarly a central limit theorem for a nonparametric estimator of the diffusi…
This document is the Part III of the book 'Stochastic Epidemic Models with Inference' edited by Tom Britton and Etienne Pardoux. Plan of the chapter: 1) Random Graphs, 2) The Reproduction Number R0, 3) SIR Epidemics on Configuration Model Graphs, 4) Statistical Description of Epidemics Spreading on Networks: The Case of Cuban HIV.
Extreme values are considered in samples with random size that has a mixed Poisson distribution being generated by a doubly stochastic Poisson process. We prove some inequalities providing bounds on the rate of convergence in limit theorems for the distributions of max-compound Cox processes.
The Neutron Transport Equation (NTE) describes the flux of neutrons over time through an inhomogeneous fissile medium. In the recent articles [5, 10], a probabilistic solution of the NTE is considered in order to demonstrate a Perron-Frobenius type growth of the solution via its projection onto an associated leading eigenfunction. In [9, 4], further analysis is performed to understand the implications of this growth both in the stochastic sense, as well as from the perspective of Monte-Carlo simulation. Such Monte-Carlo simulations are prevalent in industrial applications, in particular where …
We prove an almost sure central limit theorem on the Poisson space, which is perfectly tailored for stabilizing functionals emerging in stochastic geometry. As a consequence, we provide almost sure central limit theorems for $(i)$ the total edge length of the $k$-nearest neighbors random graph, $(ii)$ the clique count in random geometric graphs, $(iii)$ the volume of the set approximation via the Poisson-Voronoi tessellation.
Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, togethe…
Moments of density-dependent branching processes and their genealogy
École Normale Supérieure, PSL Université（巴黎高等师范学院，PSL大学）；University of Vienna（维也纳大学）；Marie et Louis Pasteur University（玛丽和路易·巴斯德大学）；Université Paris Cité（巴黎西岱大学）
We are concerned with the question of well-posedness of stochastic three dimensional incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak--strong uniqueness; (iii) non-uniqueness in law; (iv) existence of a strong Markov solution; (v) non-uniqueness of strong Markov solutions; all hold true within this class. Moreover, as a byproduct of (iii) we obtain existence and non-uniqueness of probabilistically strong and analytically weak solutions defined up to a stopping time and satisfying an energy inequality.
A celebrated result of Karpelevi\v c describes $Θ_n,$ the collection of all eigenvalues arising from the stochastic matrices of order $n.$ The boundary of $Θ_n$ consists of roots of certain one-parameter families of polynomials, and those polynomials are naturally associated with the so--called reduced Ito polynomials of Types 0, I, II and III. In this paper we explicitly characterise all $n \times n$ stochastic matrices whose characteristic polynomials are of Type 0 or Type I, and all sparsest stochastic matrices of order $n$ whose characteristic polynomials are of Type II or Type III. The re…
In Section 1, we revisit the partial differential equation (PDE) for the probability generating function (PGF) of the time-nonhomogeneous BDI (birth-and-death-with-immigration) process and derive a closed form solution. To the best of our knowledge, this is a new mathematical result. We state this result as Proposition 1. We state as Corollary 1 that the negative binomial distribution of the time-homogeneous BDI process discussed in Part I extends to the general time-nonhomogeneous case, provided that the ratio of the immigration rate to the birth rate is a constant. In section 1.2, we take up…
We study limit theorems for time-dependent averages of the form $X_t:=\frac{1}{2L(t)}\int_{-L(t)}^{L(t)} u(t, x) \, dx$, as $t\to \infty$, where $L(t)=\exp(λt)$ and $u(t, x)$ is the solution to a stochastic heat equation on $\mathbb{R}_+\times \mathbb{R}$ driven by space-time white noise with $u_0(x)=1$ for all $x\in \mathbb{R}$. We show that for $X_t$ (i) the weak law of large numbers holds when $λ>λ_1$, (ii) the strong law of large numbers holds when $λ>λ_2$, (iii) the central limit theorem holds when $λ>λ_3$, but fails when $λ<λ_4\leq λ_3$, (iv) the quantitative central limit theorem holds …
For each of (i) arbitrary stochastic reset, (ii) deterministic reset with arbitrary period, (iii) reset at arbitrary constant rate, and then in the sense of either (a) first-order stochastic dominance or (b) expectation (i.e. for each of the six possible combinations of the preceding), those laws of random times are precisely characterized that are rendered no bigger [rendered no smaller; left invariant] by all possible restart laws (within the classes (i), (ii), (iii), as the case may be). Partial results in the same vein for reset with branching are obtained. In particular it is found that d…
Nonlinear PDEs with modulated dispersion III: multiplicative noises
CNRS, Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay（法国国家科学研究中心、凡尔赛数学实验室、巴黎西岱大学、巴黎萨克雷大学）；School of Mathematics, Monash University（蒙纳士大学数学学院）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；The Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）
We present an elementary model of random size varying population given by a stationary continuous state branching process. For this model we compute the joint distribution of: the time to the most recent common ancestor, the size of the current population and the size of the population just before the most recent common ancestor (MRCA). In particular we show a natural mild bottleneck effect as the size of the population just before the MRCA is stochastically smaller than the size of the current population. We also compute the number of old families which corresponds to the number of individual…
In Part I (G.Olshanski, math.RT/9804086) and Part II (A.Borodin, math.RT/9804087) we developed an approach to certain probability distributions on the Thoma simplex. The latter has infinite dimension and is a kind of dual object for the infinite symmetric group. Our approach is based on studying the correlation functions of certain related point stochastic processes. In the present paper we consider the so-called tail point processes which describe the limit behavior of the Thoma parameters (coordinates on the Thoma simplex) with large numbers. The tail processes turn out to be stationary proc…
