décima Escuela de Probabilidad y Procesos Estocásticos

Conferencias

Expositores confirmados

Conferencias

(en orden de presentación)

Creeping of Lévy Processes through Curves, Loïc Chaumont

A Lévy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph {(t,f(t)) : t ≥ 0} of any continuous, non increasing function f such that f(0) > 0, we give an expression of the probability that a bivariate subordinator (Y,Z) issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components Y and Z. We apply this result to the creeping probability of any real Lévy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. This is a joint work with Thomas Pellas.

Fragmentation Process derived from $\alpha$-stable Galton-Watson trees, Gabriel Berzunza

Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation process derived from deleting the edges of a uniform random tree on $n$ labelled vertices. In particular, they showed that, after proper rescaling, the above fragmentation process converges as $n \rightarrow \infty$ to the fragmentation process of the Brownian CRT obtained by cutting-down the Brownian CRT along its skeleton in a Poisson manner.

In this talk, we will discuss the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree $t_n$ conditioned on having $n$ vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index $\alpha \in (1,2]$. The main result establishes that, after rescaling, the fragmentation process of $t_n$ converges, as grows to infinity, to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an $\alpha$-stable L\'evy tree. We will also explain how one can construct the latter by considering the partitions of the unit interval induced by the normalized $\alpha$-stable L\'evy excursion with a deterministic drift. In particular, the above extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.

The approach uses the well-known Prim's algorithm (or Prim-Jarník algorithm) to define a consistent exploration process that encodes the fragmentation process of $t_n$. We will discuss the key ideas of the proof.

Enumeration  of discrete trees (Complemento del Curso de Curien), Alice Contat

On Multitype Branching Processes with Interactions, Sandra Palau

Motivated by the stochastic Lotka-Volterra model, we introduce continuous-time discrete-state interacting multitype branching processes (both through intratype and intertype competition or cooperation). We show that these processes can be obtained as the sum of a multidimensional random walk with a Lamperti-type change proportional to the population size; and a multidimensional Poisson process with a time-change proportional to the pairwise interactions. We define the analogous continuous-state process as the unique strong solution of a multidimensional stochastic differential equation. Finally, we prove that a large population scaling limits of the discrete-state process correspond to its continuous counterpart. In addition, we show that the continuous-state model can be constructed as a generalized Lamperti-type transformation of multidimensional Lévy processes. Joint work with María Clara Fittipaldi.

On the geometry of large multi-conditioned random maps, Cyril Marzouk

A planar map is the embedding on the sphere of a (planar) graph, viewed up to continuous deformations. Random maps serve especially as simple models of random plane geometry and after a short overview of both the theory of local limits and scaling limits of maps, I will present some recent work on the model of maps sampled uniformly at random with a given number of vertices, edges, and faces, when all these quantities tend to infinity. We will focus especially on the « sparse » regime, in which the number of faces is small compared to the number of edges.

This is based on joint works with Nicolas Curien & Igor Kortchemski (arXiv:2101.01682 and then arXiv:2112.10719)

When is the convex minorant of a Lévy path smooth?, Jorge Ignacio González Cázares

We characterise, in terms of their transition laws, the class of one-dimensional Lévy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is exhibited by a broad class of infinite variation Lévy processes and depends subtly on the behaviour of the Lévy measure at zero. We introduce a class of strongly eroded Lévy processes, whose Dini derivatives vanish at every local minimum of the trajectory for all perturbations with a linear drift, and prove that these are precisely the processes with smooth convex hulls. We study how the smoothness of the convex hull can break, constructing examples exhibiting a variety of smooth/non-smooth behaviours. Finally, we conjecture that an infinite variation Lévy process is either strongly eroded or abrupt, a claim implied by Vigon's point-hitting conjecture. 

Forward and backward genealogies of Galton-Watson trees, Airam Blancas

A Galton-Watson process is a Markov chain which determine the total number of individuals in an asexual population, where at every discrete generation individuals give birth a random number of children, independently of the others.

In the late 90's, Le Gall and Le Jan defined the exploration process to describe, forward in time, the genealogical tree of a population evolving according with a Galton-Watson process. The backwards genealogy is given by a coalescent point process, introduced by Lambert and Popovic (2013). In this talk, we present both perspectives. Moreover, we define a Markov process to describe the genealogy of a Galton-Watson in varying environment, using the minimal amount of information. The latter process emerged to model a population where the reproduction law of the individuals is characterized by an environment.

Convergence of multitype Bienaym\'e-Galton-Watson processes conditioned on the sizes by types, Osvaldo Angtuncio

In this talk we consider a multitype Bienaymé-Galton-Watson (MBGW) forest with d types, conditioned on having ni vertices of type i for each of its types. This model is a natural generalization of unitype Bienaymé-Galton-Watson trees conditioned to have size n, which where shown by Aldous to converge to the Continuum Random Tree (1991) under a finite variance hypothesis, as n goes to infinity. First, under some general conditions, we obtain the joint law of the total number of individuals of type i in the MBGW forest. This generalizes the Otter-Dwass (1969) and Kemperman¢s formula (1950). Then, we prove that the rescaled breadth-first walk of the forest, which encodes the whole genealogy, converges to a multidimensional first-passage bridge as the number of individuals of each type goes to infinity, generalizing a result of Chaumont-Pardo (2009). Finally, we prove that the rescaled profile (the number of individuals by types at each generation) converges, which generalizes a result by Caballero-Pérez-Uribe Bravo (2017). Our results can be viewed as characteristics of a (not yet defined) object, that we call \emphmultitype Lévy forest.

Atypical rate of invasion of the reducible multitype branching Brownian motion, Bastien Mallein

A (binary) branching Brownian motion is a particle system on the real line in which particles move independently as Brownian motions, while splitting at rate 1 into (a pair of) daughter particles. We take interest in a multitype version of this process, in which the diffusion constant of the displacement and the branching rate are both influenced by the type.

When the process is irreducible (e.g. when particles of type 1 can give birth to particles of type 2, but not reciprocally), an anomalous spreading phenomenon may occur, in which the speed of the multitype process is strictly larger than the speed of each "pure" process. We take interest in the asymptotic behavior of the particles close to the maximal occupied position in this setting, showing the convergence in law of the extremal process centered around the median of the maximal displacement. This is based on a joint work with Mohamed Ali Belloum. 

Gerber-Shiu function for a class of Markov-modulated Lévy risk processes with two-sided jumps, Henry Pantí

In this work we investigate the Gerber-Shiu discounted penalty function for Markov-modulated Lévy risk processes with random incomes. Firstly, we consider the case when the downward and upward jumps (respectively, claims and random gains) are given by independent compound Poisson processes, with claim sizes with a general distribution function and gains in such a way that their distribution has a rational Laplace transform. Afterwards, we use the above results and weak convergence techniques to study the case when the claims are given by a subordinator and, subsequently, we establish results when the claims are governed by a pure jump spectrally positive Lévy process.

Joint work with Ehyter Martín-González and Antonio Murillo-Salas

Topologies for continuum measured labeled trees (Complemento del curso de Curien), Armand Riera

When k tribes go to war – a point is all that you can score, Andreas Kyprianou

A population of k murderous tribes exist. At rate C(i,j), an individual from tribe i meets with an individual from tribe j (for j = 1,….,k), killing them and reducing the j-th tribe’s by one individual. This includes infighting within tribes. Eventually, there is one surviving individual. One may think of this stochastic process as a multi-type version of Kingman’s coalescent death chain. In this talk, we investigate what happens as the total population across all tribes tends to infinity. In essence, if we describe n(t) = (n_1(t),…., n_k(t)) as the process describing the number of individuals in each tribe at time t >0, we discuss what it means for this process to “come down from infinity”. In doing so, we uncover a remarkable connection with the so-called replicator equations, describing the dynamical system of an evolutionary game theory. For this reason, we call our stochastic process the “replicator coalescent” It turns out that, from the many “infinities” the process can come down from, there is a “bottleneck” that the process must always pass through which corresponds to an evolutionary stable state in the replicator equations. Prior to this bottle neck, which occurs at arbitrarily large population size, through a time change, we see that the replicator coalescent behaves essentially like the solution to the replicator equations. After the bottleneck, stochastic effects become more pronounced and the process behaves more like a Markov chain.