We give a construction of the transition semigroup of a continuous-state branching process (CB-process) by considering the scaling limit of sequence of classical discrete-state branching processes. The transition laws of the CB-process constitute a convolution semigroup on the space of positive càdlàg paths, which defines a increasing path-valued Lévy process. The Lévy measure of the path-valued process is identified as the Kuznetsov measure associated with an entrance rule of the CB-process. We discuss a Lévy-Itô representation of the path-valued Lévy process, which gives the flow of CB-processes starting from all initial values. Another construction of the flow is given by the pathwise unique solutions to a stochastic equation driven by some time-space Lévy noise. We discuss some applications of the stochastic equation in the study of structural and distributional properties of the CB-process.

The \(N\)-particle branching random walk is a branching-selection system consisting of \(N\) particles located on the real line. At every time step, each particle is replaced by two offspring, and each offspring particle makes a jump from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the \(N\) rightmost particles survive; the other particles are removed from the system to keep the population size constant.

I will discuss results and open conjectures about the long-term behaviour of this particle system when \(N,\) the number of particles, is large. In particular, we will be interested in the asymptotic speed of the cloud of particles, and the genealogy of a sample. It turns out that the behaviour depends strongly on the tails of the jump distribution.

A critical Galton-Watson tree (which is supposed to get extinct in finite time a.s.) may be conditioned on reaching some height \(n\). Kesten has proven that this conditioned tree converges in law (with respect to the local topology) when \(n\to\infty\) toward an immortal tree. There are other ways of conditioning a tree to be large (by conditioning on the total number of nodes, the number of leaves,...). We first recall that most of the usual conditionings lead to the same limit (for a critical Galton-Watson tree). We then introduce some very specific conditionings that give rise to different limits which appeared recently with a combinatorial interpretation. We then study the analogous problem for the continuum Brownian tree.

Power fractional laws, first introduced by Sagitov and Lindo [4], form a generalization of linear fractional laws. The latter have their well-established place in the theory of branching processes because, when used to model reproduction, often lead to more explicit results than other distributions on the set of integers. This is even true in the situation of an underlying random environment as recently demonstrated in the expository article [1]. After an introduction of some fundamental properties of power fractional laws, this talk aims to explain that they form a useful extension for which explicit results can still be obtained to an extent far beyond the general case. This will be exemplified by a collection of typical examples in both fixed and i.i.d. random environment, the latter leading to a connection with affine linear recursions and perpetuities as in the linear fractional case. If time allows, I will also briefly touch on a host-parasite coevolution model due to Kimmel and Bansaye [3,2] with regard to the use of power fractional laws for the modeling of parasite multiplication. This is based on joint work with Viet Hung Hoang and Thomas Kleine Büning.

- [1] G. Alsmeyer.
*Linear fractional Galton-Watson processes in random environment and perpetuities.*Stoch. Qual. Control, 36(2):111--127, 2022. - [2] V. Bansaye.
*Proliferating parasites in dividing cells: Kimmel's branching model revisited.*Ann. Appl. Probab., 18(3):967--996, 2008. - [3] M. Kimmel.
*Quasistationarity in a branching model of division-within-division.*In Classical and modern branching processes (Minneapolis, MN, 1994), volume 84 of IMA Vol. Math. Appl., pages 157--164. Springer, New York, 1997. - [4] S. Sagitov and A. Lindo.
*A special family of Galton-Watson processes with explosions.*In Branching processes and their applications, volume 219 of Lect. Notes Stat., pages 237--254. Springer, Cham, 2016.

We consider a stochastic SIR (susceptible \(\to\) infective \(\to\) recovered) model for the spread of an epidemic on a network, described initially by an Erdös-Rényi random graph, in which susceptible individuals connected to infectious neighbours may drop or rewire such connections as a preventive measure. The early stages of an epidemic with few initial infectives may be approximated by a branching process, which enables a critical value \(\lambda_c\) of the infection rate \(\lambda\) to be determined such that, in the limit \(n \to \infty\), a major outbreak that infects at least \(\log n\) indivdiuals occurs with non-zero probability if and only if \(\lambda>\lambda_c\). A novel construction of the model is presented and used to both derive a deterministic model for epidemics started with a positive fraction initially infected and to prove convergence of the scaled stochastic model to that deterministic model as the population size \(n \to \infty\). The final size (i.e. total number of individuals infected)
of the stochastic epidemic model is also studied, focussing on epidemics initiated by few infective that take off and lead to a major outbreak. For part of the parameter space, in the limit as \(n \to \infty\), the fraction of the population infected by such a major outbreak, \(\tau(\lambda)\), is discontinuous in the infection rate \(\lambda\) at its threshold \(\lambda_c\), thus not converging to
\(0\) as \(\lambda \downarrow \lambda_c\).

Based on work done jointly with Tom Britton (Stockholm University).

We will focus on a population structured spatially by a point Poisson process on \(\mathbb{R}^2\).
We will first explain how to construct the process that combines movement, birth, death and infection, allowing for an unbounded initial condition. The rate of movement from one site to another will be strongly decreasing with distance.
The objective of the presentation will then be to obtain approximations in a diffusive regime, involving a stochastic homogenization
for the motion. In these scales, the population size will be large, the distances renormalized and the motion fast.
We could also discuss the extension to more general spatial random graphs and other scales.

Work in collaboration with Michele Salvi (Roma)

We study the asymptotic behavior of conditional least squares (CLS) estimators of drift parameters for *supercritical* continuous state and continuous time branching processes with immigration (CBI processes) based on discrete time (low frequency) observations. According to our knowledge, results on stable (mixing) convergence of CLS estimators for parameters of CBI processes
are not available in the literature, all the existing results state convergence in distribution of the appropriately normalized CLS estimators in question. For supercritical discrete time Galton-Watson branching processes, Häusler and Luschgy [3] proved
stable convergence of the CLS estimator of the offspring mean under non-extinction, which served us as a motivation for investigating the problem for CBI processes.

In case of a nontrivial immigration mechanism, under second order moment conditions on the initial law and on the branching and immigration mechanisms, we describe the asymptotic behavior of the CLS estimator of (transformed) drift parameters for a supercritical CBI process, by proving stable convergence. The limit distribution is mixed normal, except a particular case. Our results immediately yield convergence in distribution of the appropriately normalized CLS estimators in question, since stable convergence implies convergence in distribution.

The main step of our proof is to establish a stable limit theorem for a martingale associated to the supercritical CBI process in question. At this point we use a multidimensional analogue of a one-dimensional stable limit theorem due to Häusler and Luschgy [3, Chapter 7] for so called explosive stochastic processes. In fact, our multidimensional analogue may be interesting in its own right as well. If time permits, then, as special cases, we present multidimensional stable limit theorems involving multidimensional normal-, Cauchy- and stable distributions as well.

The talk is based on the papers Barczy and Pap [1] and Barczy [2].

- [1] Barczy, M., Pap, G. (2023)
*A multidimensional stable limit theorem*, Filomat 37(11), 3493--3512. - [2] Barczy, M. (2022)
*Stable convergence of conditional least squares estimators for supercritical continuous state and continuous time branching processes with immigration*Arxiv: 2207.14056. - [3] Häusler, E., Luschgy, H. (2015) Stable Convergence and Stable Limit Theorems, Springer, Cham.

I will briefly explain what the Lambda asymmetric process is in order to study its asymptotic behavior for large population and the moment duality, with emphasis in the case of two identically distributed CB processes, where duality is used to find an interesting homeomorphism between CB processes and Lambda Coalescents.

We consider multitype Galton-Watson trees where labels are non-negative integers. They generalize the simply generated trees and have appeared recently in many places in discrete random geometry: random split trees, conditioned Bienaymé-Galton-Watson trees, peeling trees in random planar maps, random fully parked trees... Under the assumption that the rescaled types along a branch converge towards a positive self-similar Markov process, those random discrete labeled trees converge in the scaling limit towards self-similar Markov branching trees (which generalized Bertoin's growth-fragmentation trees). The talk will be devoted to set the basics of this theory and to derive a few geometric consequences of the scaling limit results.

Based on a joint project with Jean Bertoin and Armand Riera.

I will discuss a joint work with Louigi Addario-Berry, in which we resolve several conjectures on the height of uniformly random trees with a given degree sequence, simply generated trees and conditioned Bienaymé-Galton-Watson trees. The proof is based on the Foata-Fuchs bijection between trees and sequences. In the second half of the talk, I will present a stochastic domination result on the height of uniformly random trees with a given degree sequence, which for example implies that binary trees are stochastically the tallest.

Consider the exchangeable fragmentation-coalescence processes with Lambda-coalescence and whose fragmentation dislocates a block at finite rate. The talk will address the following questions: When the process is started from infinitely many blocks, will it visit the partitions with finitely many blocks? (Coming down from infinity). Conversely, is the process started with finitely many blocks visiting partitions with infinitely many blocks ? (Explosion). Different parameters measuring how coalescence and fragmentation interplay will be presented and some phase transitions explained. The explosion part is based on a joint work with Xiaowen Zhou (Montreal).

A class of controlled branching processes in continuous time are considered in this talk. Given a discrete-time controlled branching process, \(\{Z_n,\ n=0,1,2\ldots\}\), and a renewal process, \(\{N_t,\ t\geq 0\}\), the process \(\{Y(t),\ t\geq 0\}\), with \(Y(t)=Z_{N(t)}\) is referred to as CBP *subordinated by a renewal process* or as *randomly indexed* CBP. We assume that the renewal period is the common lifespan of all individuals. We establish two limit theorems when the mean of the renewal periods is either finite or infinite with zero being an absorbing state. After that, we will focus our attention in studying such processes allowing an immigration component at zero. We propose an extension of the process \(\{Y(t),\ t\geq 0\}\), namely, the regenerative process \(\{U(t),\ t\geq 0\}\). It coincides with \(\{Y(t),\ t\geq 0\}\) until it hits zero, then upon staying at zero for a random time period, the process regenerates. We will provide the asymptotic behaviour of \(\{U(t),\ t\geq 0\}\). This is a joint work with M. Molina, I. del Puerto, G.P. Yanev and N.M. Yanev.
The results given in the talk have been published in González et al (2021a,b).
References:

- González, M., Molina, M., del Puerto, I.M., Yanev, G.P., Yanev, N.M. (2021a).
*Controlled branching processes with continuous time.*Journal of Applied Probability, 58 (3): 830-- 848. - del Puerto, I.M., Yanev, G.P., Molina, M., Yanev, N.M., González, M. (2021b).
*Continuous-time controlled branching processes.*Comptes Rendus de l'Académie Bulgare Des Sciences, 74 (3): 332--42

One of the open problems in the study of multitype branching processes in random environment is the construction of the associated martingale. We shall show how this construction is related to a stochastic version of the Perron-Frobenius theorem for products of random matrices. We discuss the corresponding Kesten-Stigum theorem and some potential applications.

Two classes of branching processes are commonly used to model biological populations living in restricted habitats: population-size dependent branching processes (PSDBPs) and controlled branching processes (CBPs). In this work, we develop connections between PSDBPs and CBPs. In particular, (1) we derive conditions for PSDBPs and CBPs to be equivalent and (2) we establish an upper bound on the total variation distance between non-equivalent CBPs and PSDBPs with matching first and second moments that tends to zero as the initial population size becomes large.

We illustrate our results with numerical examples.

For variable speed branching Brownian motion, it has been shown in Bovier and Hartung (2014, 2015) that the maximum position \(M_t\) of all particles alive at time \(t\), suitably centred by a deterministic function \(m_t\), converges weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as \(t\rightarrow\infty\), \( \mathbb{P}(M_t \geq \alpha m_t), \alpha>1.\) For simplicity, we only consider the two speed case that the speed is \( \sigma_1 \) for \( s \leq b t \) and \( \sigma_2 \) for \( b t < s \leq t \). For \( \sigma_1 < \sigma_2 \), the decay rate function is the same to the usual branching Brownian motion. For \( \sigma_1 > \sigma_2, \) the decay rate function exhibits a phase transition depending on a certain relation between \( \alpha \) and \( \sigma_2 \). We also give a full description of the corresponding conditioned extremal processes. The talk is based on a joint work with Xinxin Chen, Zengcai Chen and Lisa Hartung.

Branching processes naturally arise as pertinent models in a variety of applications such as population size dynamics, neutron transport and cell proliferation kinetics.
A key result for understanding the behaviour of such systems is the Perron Frobenius decomposition, which allows one to characterise the large time average behaviour of the branching process via its leading eigenvalue and corresponding left and right eigenfunctions. However, obtaining estimates of these quantities can be challenging, for example when the branching process is spatially dependent with inhomogeneous rates. In this talk, I will introduce a new interacting particle model that combines the natural branching behaviour of the underlying process with a selection and resampling mechanism, which allows one to maintain some control over the system and more efficiently estimate the eigenelements. I will then present the main result, which provides an explicit relation between the particle system and the branching process via a many-to-one formula and also quantifies the \(L^2\) distance between the occupation measures of the two systems. Finally, I will discuss some examples in order to illustrate the scope and possible extensions of the model, and to provide some comparisons with the Fleming Viot interacting particle system.

This is based on ongoing work with Alex Cox (University of Bath) and Denis Villemonais (Université de Lorraine).

The talk is based on joint works with Xinxing Chen, Victor Dagard, Bernard Derrida, Mikhail Lifshits, and Zhan Shi.

In order to study the depinning transition in presence of strong disorder, Derrida and Retaux [J. Stat. Phys. (2014)] introduced a discrete-time max-type recursive model. It is believed that for a large class of recursive models, including the Derrida and Retaux model, there are highly non-trivial universalities at and near the critical regimes. In this talk, we will review some recent results and open questions on this model.

I will discuss two SPDE models for spatial branching processes with interaction and some recently proved properties.

The first result concerns stochastic reaction diffusion equations with Feller noise. The solution to such an equation may be viewed as a spatial branching process with interaction (drift) governed by the reaction term. A natural question is if the population will go extinct in finite time almost surely. I will discuss an ongoing work with Lea Popovic in which we prove that, for a large class of reaction terms including bistable ones, there is an extinction-persistence phase transition in the branching rate: for sufficiently slow branching, solutions may survive forever; for large branch rate, they go extinct a.s.

The second result concerns solutions to a class of stochastic heat equations which may be viewed as the spatial analogues of non-linear CSBPs, or alternatively as superprocesses with density-dependent branch rate. I will discuss a recent paper which proves the compact support property for these processes in the case of stable noise. This result generalizes known results concerning superprocesses and SPDE with Gaussian noise.

Little has been written about moments higher than \(2\) for branching processes. In this talk we explore some very general results for non-local spatial branching Markov processes and non-local superprocesses alike which give straightforward limiting results under a natural Perron-Frobenius type assumptions for the first moment semigroup. The method is so robust that we are also able to establish moment growth for the occupation measure of the same classes of spatial branching processes.

We consider branching Brownian motion in which initially there is one particle at \(x\), particles produce a random number of offspring with mean \(m\) at the time of branching events, and each particle branches at rate \(\beta = 1/2m\). Particles independently move according to Brownian motion with drift \(-1\) and are killed at the origin. It is well-known that this process eventually dies out with positive probability. We condition this process to survive for an unusually large time \(t\) and study the behavior of the process at small times \(s \ll t\) using a spine decomposition. We show, in particular, that the time when a particle gets furthest from the origin is of the order \(t^{5/6}\).

The aim of this talk is to present some results on the parameter estimation on population-size-dependent branching processes (PSDBPs) whose extinction occurs with probability one.

We consider the sample defined by the population sizes and a parametric framework for the reproduction laws. In this setting, we introduce a new family of weighted least-squares estimators. We discuss the asymptotic properties of these estimators establishing their consistency and asymptotic normality. The results are illustrated by means of some simulated examples and applied to a real dataset on the black robin population.

This is a joint work with Peter Braunsteins (UNSW Sydney) and Sophie Hautphenne (University of Melbourne).

In this talk, we analyse the genealogy of a sample of \(k\) particles without replacement from a population alive at large time in a critical Galton Watson process in varying environment. Our approach uses \(k\) distinguished spine particles and a suitable change of measure similar to a \(k\)-size biasing with a discounting rate proportional to the total population size. Under this measure, we provide the law of the splitting times of the spines and the offspring distribution of particles on and off the spines

We study the asymptotic behavior of the probability of non extinction of a weakly subcritical multitype branching process in iid random environment. Under suitable and quite general assumptions, the survival probability is of order of \(r^n n^{-3/2}\)
for some \(r\) to specify.

Joint work with D.C. Pham.

In the 70's Itô settled the excursion theory of Markov processes, which is nowadays a fundamental tool for analyzing path properties of Markov processes. In his theory, Itô also introduced a method for building Markov processes using the excursion data, or by gluing excursions together, the resulting process is known as the recurrent extension of a given process. Since Itô's pioneering work the method of recurrent extensions has been added to the toolbox for building processes, which of course includes the martingale problem and stochastic differential equations. The latter are among the most popular tools for defining stochastic processes, in particular in applied models as they allow to physically describe the infinitesimal variations of the studied phenomena. In this work we answer the following natural question. Assume \(X\) is a Markov process taking values in \(\mathbb{R}\), that dies at the first time it hits a distinguished point of the state space, say \(x_0\in\mathbb{R}\), which happens in a finite time a.s. that \(X\) satisfies a stochastic differential equation, and finally that \(X\) admits a recurrent extension, say \(\widetilde{X}\), that is a processes that behaves like \(X\) up to the first hitting time of \(x_0\), and for which \(x_0\) is a recurrent and regular state. If any, what is the SDE satisfied by \(\widetilde{X}\) ? Our answer allows us to describe the SDE satisfied by many Feller processes. We analyze various particular examples, as for instance the so-called Feller brownian motions and diffusions, which include their sticky and skewed versions, and also self-similar Markov processes, continuous state branching processes and real valued Lévy processes.

Consider a birth-death process started from one individual in which each individual gives birth at rate \(\lambda\) and dies at rate \(\mu\), so that the population size grows at rate \(r = \lambda - \mu\). Lambert (2018) and Harris, Johnston, and Roberts (2020) came up with methods for constructing the exact genealogy of a sample of size \(n\) taken from this population at time \(T\). We use the construction of Lambert (2018), which is based on the coalescent point process, to obtain asymptotic results for the site frequency spectrum associated with this sample. We also explain how to apply these results to obtain a confidence interval for the growth rate of an exponentially growing tumor.

The continuum Derrida-Retaux system originates from some hierarchical renormalization models in statistical physics, and can be viewed as an exactly solvable growth-fragmentation process in the sense of Bertoin. I am going to make elementary discussions on asymptotic properties of the system.

Joint work with Bernard Derrida and Thomas Duquesne.

We will introduce and study a variant of a well-known model in population genetics, named Muller's ratchet, which is seen as one explanation of the ubiquity of sexual selection in Nature. Consider a population of \(N\) individuals, each of them carrying a type
in \(\mathbb{N}\). The population evolves according to a Moran dynamics with selection and mutation, where an individual of type \(k\) has the same selective advantage over all individuals with type \(k'>k\), and type \(k\) mutates to type \(k+1\) at a constant rate (in the classical Muller's ratchet, the selective advantage is proportional to \(k'−k\) ). For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the click times of the ratchet (i.e. the times at which the hitherto minimal (`best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a `dual' hierarchical multitype logistic system. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations.

This is a joint work with A. Gonzalez Casanova and Anton Wakolbinger.

The CRT is the scaling limit of the UST on the complete graph. The Aldous Broder chain on a graph \(G=(V,E)\) is a MC with values in the space of rooted trees with vertices in \(V\) that is invariant under the uniform distribution on the space of rooted trees spanning
\(G\). In Evans, Pitman and Winter (2006) the so-called root growth with regrafting process (RGRG) was constructed. It was further shown that the suitable rescaled Aldous Broder chain converges to the RGRG weakly with respect to the GH-topology. It was shown in Peres and Revelle (2005) that (up to a dimension depending constant factor) the CRT is also the Gromov-weak scaling limit of the UST on the \(d\)-dimensional torus, \(d \geq 5\). This result was recently strengthens in Archer, Nachmias and Shalev (2021+) to convergence with respect to the GH-weak topology, and therefore also with respect to the GH-topology. In this talk we show that also the suitable rescaled Aldous Broder chain converges to the RGRG weakly with respect to the GH-topology when initially started in the trivial rooted tree.

Joint work with Osvaldo Angtuncio Hernandez and Gabriel Berzunza Ojeda.

Continuous-state branching processes (CSBPs) with nonlinear branching mechanism can be obtained from spectrally positive Lévy processes by generalized Lamperti time change. These generalized CSBPs allow rich behaviours of extinction, explosion and coming down from infinity. The explosion behaviours for nonlinear CSBPs have been studied in Li and Zhou (2021) when the big jumps of the process have a finite first moment. In this talk we further consider the explosion behaviours for such processes with jumps of infinite first moment. In particular, we identify the speed of explosion when the associated Laplace exponent and rate function are both regularly varying.

This talk is based on joint work with Clement Foucart and Bo Li.