arXiv subjects

Clément Foucart

Publications and source records attributed to Clément Foucart.

At least 19 recordsLinked to original sources

arXiv2026

Continuous-state branching processes with Lévy-Khintchine drift-interaction: Laplace duality and Fellerian extensions

We investigate the class of continuous-state branching processes with interaction driven by a Lévy-Khintchine type drift (CBDI). These $[0,\infty]$-valued processes capture both dynamics of branching and density-dependence, allowing for cooperation at low population sizes and competition at high densities. Although the interaction breaks the branching property, the Lévy--Khintchine form of the drift induces a Laplace duality. This duality expresses the Laplace transform of a CBDI process in terms of that of another CBDI process, in which the branching and drift-interaction mechanisms are exchanged. The process, stopped upon hitting either boundary $0$ or $\infty$, is uniquely characterized in law by these mechanisms. A Fellerian extension is constructed when the drift is non-Lipschitz and sufficiently strong at a boundary, allowing the process to leave this boundary continuously and possibly re-enter it. We identify parameters, defined in terms of the mechanisms and their associated scale function and potential measure, that determine the boundary behavior at $0$ and $\infty$ (entrance, exit or regular). Settings exhibiting all regimes, including regular-for-itself and non-sticky boundaries, arise when the mechanisms are assumed to be regularly varying. Our approach combines Laplace duality and comparison principles. The duality facilitates the analysis of semigroups and the construction of sharp Lyapunov functions. Comparisons ensure monotonicity and convergence properties of first-passage times.

math.PR↗

arXiv2026

Positive Markov processes in Laplace duality

This article develops a general framework for Laplace duality between positive Markov processes in which the one-dimensional Laplace transform of one process can be represented through that of another. We show that a process admits a Laplace dual if and only if it satisfies a certain complete monotonicity condition. Moreover, we analyse how the conventions adopted for the values of $0 \cdot \infty$ and $\infty \cdot 0$ are reflected in the weak continuity/absorptivity properties of the processes in duality at the boundaries $0$ and $\infty$. A broad class of generators admitting Laplace duals is identified, and we provide sufficient conditions under which the associated martingale problems are well-posed with the solutions being in duality at the level of their semigroups. Laplace duality is shown to furnish a unifying structure for several generalizations of continuous-state branching processes, e.g. those with immigration or evolving in random environments. Along the way, a theorem originally due to Ethier and Kurtz -- connecting duality of generators to that of the associated semigroups -- is refined, and we provide a concise proof of the Courrège form for the pointwise infinitesimal generator of a positive Markov process whose domain includes the exponential functions. The latter leads naturally to the notion of a Laplace symbol, which is a parsimonious encoding of the infinitesimal dynamics of the process.

math.PR↗

arXiv2025

Local explosions and extinction in continuous-state branching processes with logistic competition

We study by duality methods the extinction and explosion times of continuous-state branching processes with logistic competition (LCSBPs) and identify the local time at $\infty$ of the process when it is instantaneously reflected at $\infty$. The main idea is to introduce a certain "bidual" process $V$ of the LCSBP $Z$. The latter is the Siegmund dual process of the process $U$, that was introduced in Foucart (2019), as the Laplace dual of $Z$. By using both dualities, we shall relate local explosions and the extinction of $Z$ to local extinctions and the explosion of the process $V$. The process $V$ being a one-dimensional diffusion on $[0,\infty]$, many results on diffusions can be used and transfered to $Z$. A concise study of Siegmund duality for one-dimensional diffusions and their boundaries is also provided.

math.PR↗

arXiv2025

Decompositions of CSBPs via Poissonian Intertwining

We revisit certain decompositions of continuous-state branching processes (CSBPs), commonly referred to as skeletal decompositions, through the lens of intertwining of semi-groups. Precisely, we associate to a CSBP $X$ with branching mechanism $ψ$ a family of $\mathbb{R}_+\times \mathbb{Z}_+$-valued branching processes $(X^λ, L^λ)$, indexed by a parameter $λ\in (0, \infty)$, that satisfies an intertwining relationship with $X$ through the Poisson kernel with parameter $λx$. The continuous component $X^λ$ has the same law as $X$, while the discrete component $L^λ$, conditionally on $X^λ_t$, has a Poisson distribution with parameter $λX^λ_t$. The law of $(X^λ, L^λ)$ depends on the position of $λ$ within $[0, \infty) = [0, ρ) \cup [ρ, \infty)$, where $ρ$ is the largest positive root of $ψ$. When $λ\geq ρ$, various well-known results concerning skeleton decompositions are recovered. In the supercritical case ($ρ> 0$), when $λ<ρ$, a novel phenomenon arises: a birth term appears in the skeleton, corresponding to a one-unit proportional immigration from the continuous to the discrete component. Along the way, the class of continuous-time branching processes taking values in $\mathbb{R}_+ \times \mathbb{Z}_+$ is constructed.

math.PR↗

arXiv2024

Extremal shot noise processes and random cutout sets

We study some fundamental properties, such as the transience, the recurrence, the first passage times and the zero-set of a certain type of sawtooth Markov processes, called extremal shot noise processes. The sets of zeros of the latter are Mandelbrot's random cutout sets, i.e. the sets of points left uncovered after placing Poisson random covering intervals on the positive half-line. Based on this connection, we provide a new proof of Fitzsimmons-Fristedt-Shepp Theorem which characterizes the random cutout sets.

math.PR↗

arXiv2024

Weak convergence of continuous-state branching processes with large immigration

Functional limit theorems are established for continuous-state branching processes with immigration (CBIs), where the reproduction laws have finite first moments and the immigration laws exhibit large tails. Different regimes of immigration are identified, leading to limiting processes that are either subordinators, CBIs, extremal processes, or extremal shot noise processes.

math.PR↗

arXiv2024

Conditioning the logistic continuous-state branching process on non-extinction via its total progeny

The problem of conditioning a continuous-state branching process with quadratic competition (logistic CB process) on non-extinction is investigated. We first establish that non-extinction is equivalent to the total progeny of the population being infinite. The conditioning we propose is then designed by requiring the total progeny to exceed arbitrarily large exponential random variables. This is related to a Doob's $h$-transform with an explicit excessive function $h$. The $h$-transformed process, i.e. the conditioned process, is shown to have a finite lifetime almost surely (it is either killed or it explodes continuously). When starting from positive values, the conditioned process is furthermore characterized, up to its lifetime, as the solution to a certain stochastic equation with jumps. The latter superposes the dynamics of the initial logistic CB process with an additional density-dependent immigration term. Last, it is established that the conditioned process can be starting from zero. Key tools employed are a representation of the logistic CB process through a time-changed generalized Ornstein-Uhlenbeck process, as well as Laplace and Siegmund duality relationships with auxiliary diffusion processes.

math.PR↗

arXiv2024

Behaviors near explosion of nonlinear CSBPs with regularly varying mechanisms

We study the explosion phenomenon of nonlinear continuous-state branching processes (nonlinear CSBPs). First an explicit integral test for explosion is designed when the rate function does not increase too fast. We then exhibit three different regimes of explosion when the branching mechanism and the rate function are regularly varying respectively at $0$ and $\infty$ with indices $α$ and $β$ such that $0\leqα\leq β$ and $β>0$. If $α>0$ then the renormalisation of the process before its explosion is linear. When moreover $α\neq β$, the limiting distribution is that of a ratio of two independent random variables whose laws are identified. When $α=β$, the limiting random variable shrinks to a constant. Last, when $α=0$, i.e. the branching mechanism is slowly varying at $0$, the process is studied with the help of a nonlinear renormalisation. The limiting distribution becomes the inverse uniform distribution. This complements results known in the case of finite mean and provides new insight for the classical explosive continuous-state branching processes (for which $β=1$).

math.PR↗

arXiv2023

Continuous-state branching processes with collisions: first passage times and duality

We introduce a class of one-dimensional positive Markov processes generalizing continuous-state branching processes (CBs), by taking into account a phenomenon of random collisions. Besides branching, characterized by a general mechanism $Ψ$, at a constant rate in time two particles are sampled uniformly in the population, collide and leave a mass of particles governed by a (sub)critical mechanism $Σ$. Such CB processes with collisions (CBCs) are shown to be the only Feller processes without negative jumps satisfying a Laplace duality relationship with one-dimensional diffusions on the half-line. This generalizes the duality observed for logistic CBs by Foucart. Via time-change, CBCs are also related to an auxiliary class of Markov processes, called CB processes with spectrally positive migration (CBMs), recently introduced by Vidmar. We find necessary and sufficient conditions for the boundaries $0$ or $\infty$ to be attracting and for a limiting distribution to exist. The Laplace transform of the latter is provided. Under the assumption that the CBC process does not explode, the Laplace transforms of the first passage times below arbitrary levels are represented with the help of the solution of a second-order differential equation, whose coefficients express in terms of the Lévy-Khintchine functions $Σ$ and $Ψ$. Sufficient conditions for non-explosion are given.

math.PR↗

arXiv2021

Limit theorems for continuous-state branching processes with immigration

We prove and extend some results stated by Mark Pinsky: Limit theorems for continuous state branching processes with immigration [Bull. Amer. Math. Soc. 78(1972), 242--244]. Consider a continuous-state branching process with immigration $(Y_t,t\geq 0)$ with branching mechanism $Ψ$ and immigration mechanism $Φ$ (CBI$(Ψ,Φ)$ for short). We shed some light on two different asymptotic regimes occurring when $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du<\infty$ or $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du=\infty$. We first observe that when $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du<\infty$, supercritical CBIs have a growth rate dictated by the branching dynamics, namely there is a renormalization $τ(t)$, only depending on $Ψ$, such that $(τ(t)Y_t,t\geq 0)$ converges almost-surely to a finite random variable. When $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du=\infty$, it is shown that the immigration overwhelms the branching dynamics and that no linear renormalization of the process can exist. Asymptotics in the second regime are studied in details for all non-critical CBI processes via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are then exhibited, where a misprint in Pinsky's paper is corrected. CBI processes with critical branching mechanisms subject to a regular variation assumption are also studied.

math.PR↗

arXiv2021

A phase transition in the coming down from infinity of simple exchangeable fragmentation-coagulation processes

We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a $Λ$-coalescent, and fragmentation dislocates at finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes, simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters $θ_{\star}\leq θ^{\star}\in [0,\infty]$, so that if $θ^{\star}<1$, the process comes down from infinity and if $θ_\star>1$, then it stays infinite. We illustrate our result with regularly varying coagulation and fragmentation measures. In this case, the parameters $θ_{\star},θ^{\star}$ coincide and are explicit.

math.PR↗

arXiv2020

On the explosion of the number of fragments in the simple exchangeable fragmentation-coalescence processes

We consider the exchangeable fragmentation-coagulation (EFC) processes, where the coagulations are multiple and not simultaneous, as in a $Λ$-coalescent, and the fragmentations dislocate at finite rate an individual block into sub-blocks of infinite size. Sufficient conditions are found for the block-counting process to explode (i.e. to reach $\infty$) or not and for infinity to be an exit boundary or an entrance boundary. In a case of regularly varying fragmentation and coagulation mechanisms, we find regimes where the boundary $\infty$ can be either an exit, an entrance or a regular boundary. In the latter regular case, the EFC process leaves instantaneously the set of partitions with an infinite number of blocks and returns to it immediately. Proofs are based on a new sufficient condition of explosion for positive continuous-time Markov chains, which is of independent interest.

math.PR↗

arXiv2020

On the boundary classification of $Λ$-Wright-Fisher processes with frequency-dependent selection

We construct extensions of the pure-jump $Λ$-Wright-Fisher processes with frequency-dependent selection ($Λ$-WF processes with selection) beyond their first passage time at the boundary $1$. We show that they satisfy some duality relationships with the block counting process of simple exchangeable fragmentation-coalescence processes (EFC). One-to-one correspondences between the nature of the boundary $1$ of the $Λ$-WF process with selection and the boundary $\infty$ of the block counting process are established. New properties for the $Λ$-WF processes with selection and the block counting processes of the simple EFC processes are deduced from these correspondences. Some conditions are provided for the selection to be either weak enough for boundary $1$ to be an exit boundary or strong enough for $1$ to be an entrance boundary. When the measure $Λ$ and the selection mechanism satisfy some regular variation properties, conditions are found in order that the extended $Λ$-WF process with selection makes excursions out from the boundary $1$ before getting absorbed at $0$. In the latter process, $1$ is a transient regular reflecting boundary. This corresponds to a new phenomenon for the deleterious allele which can spread into the population in a set of times of zero Lebesgue measure, before vanishing in finite time almost surely.

math.PR↗

arXiv2020

Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations

Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). Individuals reproduce independently according to the same subcritical offspring distribution. We study the long-term behaviour of the ancestral lineages as time goes to the past and show that the flow of ancestral lineages, properly renormalized, converges almost surely to the inverse of a drift-free subordinator whose Laplace exponent is explicit in terms of the branching mechanism. We provide an interpretation in terms of the genealogy of the population. In particular, we show that the inverse subordinator is partitioning the current population into ancestral families with distinct common ancestors. When Grey's condition is satisfied, the population comes from a discrete set of ancestors and the ancestral families are i.i.d and distributed according to the quasi-stationary distribution of the CSBP conditioned on non-extinction. When Grey's condition is not satisfied, the population comes from a continuum of ancestors which is described as the set of increase points $\mathscr{S}$ of the limiting inverse subordinator. The Hausdorff dimension of $\mathscr{S}$ is given. The proof is based on a general result for stochastically monotone processes of independent interest, which relates $θ$-invariant measures and $θ$-invariant functions for a process and its Siegmund dual.

math.PR↗

arXiv2020

Time-changed spectrally positive Lévy processes starting from infinity

Consider a spectrally positive Lévy process $Z$ with log-Laplace exponent $Ψ$ and a positive continuous function $R$ on $(0,\infty)$. We investigate the entrance from $\infty$ of the process $X$ obtained by changing time in $Z$ with the inverse of the additive functional $η(t)=\int_{0}^{t}\frac{{\rm d} s}{R(Z_s)}$. We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process $X$. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the Lévy process has a negative drift $δ:=-γ<0$, sufficient conditions over $R$ and $Ψ$ are found for the process to come down from infinity along the deterministic function $(x_t,t\geq 0)$ solution to ${\rm d} x_t=-γR(x_t) {\rm d} t$, with $x_0=\infty$. When $Ψ(λ)\sim cλ^α$, with $λ\rightarrow 0$, $α\in (1,2]$, $c>0$ and $R$ is regularly varying at $\infty$ with index $θ>α$, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.

math.PR↗

arXiv2020

On the entrance at infinity of Feller processes with no negative jumps

Consider a non-explosive positive Feller process with no negative jumps. It is shown in this note that when infinity is an entrance boundary, in the sense that the entrance times of the process remain bounded when the initial value tends to infinity, the process admits a Feller extension on the compactified state space $[0,\infty]$. Moreover, when started from infinity, the extended Markov process on $[0,\infty]$ leaves infinity instantaneously and stays finite, almost-surely. Arguments are adapted from a proof given by O. Kallenberg for diffusions. We also show that the process started from $x$ converges weakly towards that started from infinity in the Skorokhod space, when $x$ goes to infinity.

math.PR↗

arXiv2018

Coalescences in Continuous-State Branching Processes

Consider a continuous-state branching population constructed as a flow of nested subordinators. Inverting the subordinators and reversing time give rise to a flow of coalescing Markov processes (with negative jumps) which correspond to the ancestral lineages of individuals in the current generation. The process of the ancestral lineage of a fixed individual is the Siegmund dual process of the continuous-state branching process. We study its semi-group, its long-term behavior and its generator. In order to follow the coalescences in the ancestral lineages and to describe the backward genealogy of the population, we define non-exchangeable Markovian coalescent processes obtained by sampling independent Poisson arrival times over the flow. These coalescent processes are called consecutive coalescents, as only consecutive blocks can merge. They are characterized in law by finite measures on $\mathbb{N}$ which can be thought as the offspring distributions of some inhomogeneous immortal Galton-Watson processes forward in time.

math.PR↗

arXiv2018

Continuous-state branching processes with competition: duality and reflection at Infinity

The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for $\infty$ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty$ is inaccessible, it is always an entrance boundary. In the case where $\infty$ is accessible, explosion can occur either by a single jump to $\infty$ (the process at $z$ jumps to $\infty$ at rate $λz$ for some $λ>0$) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when $\infty$ is accessible and $0\leq \frac{2λ}{c}<1$, the extended process is reflected at $\infty$. In the case $\frac{2λ}{c}\geq 1$, $\infty$ is an exit of the extended process. When the branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at $\infty$ get extinct almost-surely. Moreover absorption at $0$ is almost-sure if and only if Grey's condition is satisfied. When the branching mechanism is the Laplace exponent of a subordinator, necessary and sufficient conditions are given for a stationary distribution to exist. The Laplace transform of the latter is provided. The study is based on classical time-change arguments and on a new duality method relating logistic CSBPs with certain generalized Feller diffusions.

math.PR↗