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Grégoire Véchambre

Publications and source records attributed to Grégoire Véchambre.

13 recordsLinked to original sources
arXiv2026

Stochastic flows and Poisson representations for the block masses of the $Λ$-coalescent with dust : moment asymptotics and large deviation estimates

We develop a new methodology for the study of the $Λ$-coalescent with dust, based on the construction of a stochastic flow of inverses, introduced by Bertoin and Le Gall [Ann. inst. Henri Poincare (B) Probab. Stat. 41(3), 307-333 (2003)], in which the coalescent is naturally embedded as a nested interval-partition. This framework yields Poisson representations for the ordered block masses $(W_k(t))_{k \geq 1}$ as stochastic integrals with respect to the Poisson random measure governing the flow, enabling the use of stochastic calculus in a setting where it was not previously available. We believe this methodology to be of independent interest and applicable beyond the specific results of this paper. As a first application, we derive precise logarithmic asymptotics for the moments of $W_k(t)$ as $t \to \infty$, which reveal an interesting cutoff phenomenon related to the presence of dust. We then establish a law of large numbers and a large deviation principle for $\log(1-W_1(t))/t$ and a one-sided weak large deviation principle for $\log(W_k(t))/t$, for $k \geq 2$, with explicit rate functions.

math.PR↗
arXiv2026

The leftmost particle of branching subordinators

We define a family of continuous-time branching particle systems on the non-negative real line, called branching subordinators, where particles move as independent subordinators. Each particle can also split (at possibly infinite rate) into several children (possibly infinitely many) whose positions relative to the position of the parent are random. These particle systems are in the continuity of branching Lévy processes introduced by Bertoin and Mallein [Ann. Probab. 47(3): 1619-1652 (2019)]. We pay a particular attention to the asymptotic behavior of the leftmost particle of branching subordinators. It turns out that, under some assumptions, the rate of growth of the position of the leftmost particle is of order $t^γ$ where $γ\in (0,1)$ depends explicitly on the parameters. This sub-linear growth is significantly different from the classical linear growth observed for regular branching random walks with non-negative displacements.

math.PR↗
arXiv2025

Large deviation principle for the absorption time of the Beta-coalescent via integral functionals

We study some aspects of the absorption time of the Beta$(a,b)$-Coalescent starting with $n$ blocks. More precisely, when $a>1$, the absorption time is known to converge to infinity as $n$ goes to infinity, and we prove that it satisfies a large deviation principle. When $a \in (0,1)$, it is known that the coalescent comes down from infinity, and we derive bounds for the convergence in the Kolmogorov distance of the distribution of the absorption time as $n$ goes to infinity. To prove our results we introduce a method, inspired from statistical mechanics, that allows to infer the asymptotic behavior of the Laplace transforms of some integral functionals of the Beta-coalescent as the initial number of blocks $n$ goes to infinity. As a by-product of our proofs we also obtain estimates for the record probabilities of the Beta-coalescent.

math.PR↗
arXiv2025

Distance to stationarity and open set recurrence for ergodic processes on the unit interval with a Bernstein or a Siegmund dual

We use duality techniques - specifically Siegmund and Bernstein duality - as tools to analyse ergodic and recurrence properties of $[0,1]$-valued Markov processes. These dualities enable the derivation of sharp bounds on the distance to stationarity and allow for a novel approach to establishing topological recurrence, whereby ergodicity implies recurrent visits to neighbourhoods of points in the stationary distribution's support. We first relate Siegmund and Bernstein duality in cases where both apply, such as $Λ$-Wright--Fisher processes. We then exploit the dual processes to derive bounds on the distance to stationarity and simple criteria for open set recurrence. Our results apply to a broad class of $Λ$-Wright--Fisher processes with frequency-dependent selection, mutation, and random environmental effects. In many cases, Bernstein duals can be constructed from the same ancestral structures underlying moment duals, allowing our methods to extend and strengthen existing results based on moment duality. This framework provides a robust approach to studying the long-term behaviour of complex population models beyond the classical moment dual setting.

math.PR↗
arXiv2024

Bernstein duality revisited: frequency-dependent selection, coordinated mutation and opposing environments

This paper investigates the long-term behavior of a class of $Λ$-Wright--Fisher processes incorporating frequency-dependent selection, coordinated (bidirectional) selection, as well as individual and coordinated mutation. Our primary analytical tool is Bernstein duality, a generalization of moment duality. We introduce the corresponding dual process and establish the relevant duality relation. Without mutation, this work complements earlier studies that employed moment duality, Siegmund duality or other methods to classify the long-term behavior of similar processes. Notably, the current analysis encompasses parameter regimes that model bidirectional selection, a scenario that has proven challenging to analyze using moment duality. In the presence of mutation, we establish the ergodic properties of the process.

math.PR↗
arXiv2024

$Λ$-Wright--Fisher processes with general selection and opposing environmental effects: fixation and coexistence

Our results characterize the long-term behavior for a broad class of $Λ$-Wright--Fisher processes with frequency-dependent and environmental selection. In particular, we reveal a rich variety of parameter-dependent behaviors and provide explicit criteria to discriminate between them. That includes the situation in which the (entire) boundary is repelling -- a new phenomenon in this context. This has significant biological implications, because it means that selection alone can maintain coexistence. If a boundary point is attractive, we derive polynomial/exponential decay rates for the probability of not being polynomially/exponentially close to that boundary, depending on some weak/strong integrability conditions. Moreover, we provide a handy representation of the fixation probability. In our proofs we make use of Siegmund duality. The dual process can be sandwiched near the boundaries in between transformed Lévy processes. In this way we relate the boundary behavior of the dual process to fluctuation properties of these Lévy processes and shed new light on previously established conditions for attractive/repelling boundary points. Our method allows us to treat models that so far could not be analyzed by means of moment or Bernstein duality. This closes an existing gap in the literature.

math.PR↗
arXiv2023

Spectral analysis of a class of Lévy-type processes and connection with some spin systems

We consider a class of Lévy-type processes on which spectral analysis technics can be made to produce optimal results, in particular for the decay rate of their survival probability and for the spectral gap of their ground state transform. This class is defined by killed symmetric Lévy processes under general random time-changes satisfying some integrability assumptions. Our results reveal a connection between those processes and a family of spin systems. This connection, where the free energy and correlations play an essential role, is, up to our knowledge, new, and relates some key properties of objects from the two families. When the underlying Lévy process is a Brownian motion, the associated spin system turns out to have interactions of a rather nice form that are a natural alternative to the quadratic interactions from the lattice Gaussian Free Field. More general Lévy processes give rise to more general interactions in the associated spin systems.

math.PR↗
arXiv2022

Combinatorics of ancestral lines for a Wright-Fisher diffusion with selection in a Lévy environment

Wright-Fisher diffusions describe the evolution of the type composition of an infinite haploid population with two types (say type $0$ and type $1$) subject to neutral reproductions, and possibly selection and mutations. In the present paper we study a Wright-Fisher diffusion in a Lévy environment that gives a selective advantage to sometimes one type, sometimes the other. Classical methods using the Ancestral Selection Graph (ASG) fail in the study of this model because of the complexity, resulting from the two-sided selection, of the structure of the information contained in the ASG. We propose a new method that consists in encoding the relevant combinatorics of the ASG into a function. We show that the expectations of the coefficients of this function form a (non-stochastic) semigroup and deduce that they satisfy a linear system of differential equations. As a result we obtain a series representation for the fixation probability $h(x)$ (where $x$ is the initial proportion of individuals of type $0$ in the population) as an infinite sum of polynomials whose coefficients satisfy explicit linear relations. Our approach then allows to derive Taylor expansions at every order for $h(x)$ near $x=0$ and to obtain an explicit recursion formula for the coefficients.

math.PR↗
arXiv2022

Moran models and Wright--Fisher diffusions with selection and mutation in a one-sided random environment

Consider a two-type Moran population of size $N$ with selection and mutation, where the selective advantage of the fit individuals is amplified at extreme environmental conditions. Assume selection and mutation are weak with respect to $N$, and extreme environmental conditions rarely occur. We show that, as $N\to\infty$, the type frequency process with time speed up by $N$ converges to the solution of a Wright-Fisher-type SDE with a jump term modeling the effect of the environment. We use an extension of the \emph{ancestral selection graph} (ASG) to describe the model's genealogical picture. Next, we show that the type frequency process and the line-counting process of a pruned version of the ASG satisfy a moment duality. This relation yields a characterization of the asymptotic type distribution. We characterize the ancestral type distribution using an alternative pruning of the ASG. Most of our results are stated in annealed and quenched form.

math.PR↗
arXiv2019

Almost sure behavior for the local time of a diffusion in a spectrally negative Lévy environment

We study the almost sure asymptotic behavior of the supremum of the local time for a transient diffusion in a spectrally negative Lévy environment. More precisely, we provide the proper renormalizations for the extremely large and the extremely small values of the supremum of the local time. In particular, in the sub-ballistic regime, we link the almost sure behavior of the supremum of the local time with the left asymptotic tail of the exponential functional of the environment conditioned to stay positive, which allows to use the properties of such exponential functionals to characterize the sought behavior. It appears from our results that the renormalization of the extremely large values of the supremum of the local time in the sub-ballistic regime is determined by the asymptotic behavior of the Laplace exponent of the Lévy environment, and is surprisingly greater than the renormalization that was previously known for the recurrent case. For the renormalization of the extremely small values of the supremum of the local time in the sub-ballistic regime, our results show that there is only one possible renormalization, whatever is the choice of the Lévy environment.

math.PR↗
arXiv2019

General self-similarity properties for Markov processes and exponential functionals of L{é}vy processes

Positive self-similar Markov processes (pssMp) are positive Markov processes that satisfy the scaling property and it is known that they can be represented as the exponential of a time-changed Lévy process via Lamperti representation. In this work, we are interested in the following problem: what happens if we consider Markov processes in dimension $1$ or $2$ that satisfy self-similarity properties of a more general form than a scaling property ? Can they all be represented as a function of a time-changed Lévy process ? If not, how can Lamperti representation be generalized ? We show that, not surprisingly, a Markovian process in dimension $1$ that satisfies self-similarity properties of a general form can indeed be represented as a function of a time-changed Lévy process, which shows some kind of universality for the classical Lamperti representation in dimension $1$. However, and this is our main result, we show that a Markovian process in dimension $2$ that satisfies self-similarity properties of a general form is represented as a function of a time-changed exponential functional of a bivariate Lévy process, and processes that can be represented as a function of a time-changed Lévy process form a strict subclass. This shows that the classical Lamperti representation is not universal in dimension $2$. We briefly discuss the complications that occur in higher dimensions. In dimension $2$ we present an example, built from a self-similar fragmentation process, where our representation in term of an exponential functional of a bivariate Lévy process appears naturally and has a nice interpretation in term of the self-similar fragmentation process.

math.PR↗
arXiv2018

Path decomposition of a spectrally negative Lévy process, and local time of a diffusion in this environment

We study the convergence in distribution of the supremum of the local time and of the favorite site for a transient diffusion in a spectrally negative Lévy potential. To do so, we study the h-valleys of a spectrally negative Lévy process, and we prove in partiular that the renormalized sequence of the h-minima converges to the jumping times sequence of a standard Poisson process.

math.PR↗