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.

 

--Marco Antonio López Ortiz: "Targeted cutting of random recursive trees"

We propose a method of cutting down a random recursive tree that focuses on its higher degree vertices. Enumerate the vertices of a random recursive tree of size $n$ according to a decreasing order of their degrees. The targeted vertex-cutting process is performed by sequentially removing the vertices in the listed order and keeping only the subtree containing the root. The algorithm ends when the root is removed.   The total number of steps for this procedure,$K_n$ , is upper bounded by $Z_{≥D}$, which denotes the number of vertices that have degree at least as large as the degree of the root. We obtain that the first order growth of $K_n$ is upper bounded by $n {1−ln 2}$, which is substantially smaller than the required number of removals if, instead, the vertices were selected uniformly at random. More precisely, we prove that $ln(Z_{≥D})$ grows as $ln(n)$ asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the k-th moment of $ln(Z_{≥D})$ is proportional to $(ln(n))^k$. 

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--Isaac González: "Asymptotic moments of spatial branching processes"

Consider either a branching Markov process or a superprocess on a general space, with non-local branching mechanism. For a general setting in which the first moment semigroup displays a Perron-Frobenius type behaviour and with an appropriate deterministic normalisation, we find the limit behaviour of the k-th order moment of the process in terms of the principal right eigenfunction and left eigenmeasure, which can be identified explicitly as either polynomial in t or exponential in t, depending on whether the process is critical, supercritical or subcritical. The method we employ is extremely robust and we are able to extract similarly precise results that additionally give us the moment growth with time of the integrated moments, which we can think of as characterising the running occupation measure of the process.

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--Carlo Scali: "Quenched invariance principle for biased random walks in random conductances in the sub-ballistic regime"

We consider a biased random walk in positive random conductances on $\mathbb{Z}^d$ for $d\geq 5$. In the sub-ballistic regime, Fribergh and Kious proved the convergence, under the annealed law, of the properly rescaled random walk towards a Fractional Kinetics. I will explain that a quenched equivalent of this theorem is true and a strategy to simplify the question. This is joint work with A. Fribergh and T. Lions.

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--Jesus Contreras: "Contributions on local times of Spectrally Negative Lévy Processes"

This talk includes two topics. The first one is about scale functions, which are well known and appear in the expression of the Laplace transform of the first exit time T of an interval. It turns out we can consider the exit problem for more general functionals related to T, the process and its supremum. We achieve this by using excursion theory, and the corresponding generalized scale functions are closely related to the process of local times via the occupation density formula. For the second part, I will briefly mention some advances we have made on trying to establish analogues of the First and Second Ray-Knight theorems for SNLPs. Both of these are joint works with Victor Rivero.

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--William Da Silva: "Growth-fragmentation processes with types"

We introduce a new family of processes with types, which are a natural extension of Bertoin's growth-fragmentation processes. We show how the theory of Markov additive processes can be leveraged to describe these processes, and study their properties. This is joint work with Juan Carlos Pardo (CIMAT).

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--Sonny Medina: "Directional excursions of the isotropic α-stable Lévy and a new example of Deep Wiener-Höpf factorisation"

In this talk we will show some fluctuation identities for the isotropic α-stable Lévy process in regard to half-spaces and their normal vectors. These complement the recent results of Kyprianou, Rivero and Satitkanitkul, in which a theory of radial excursions of this process is developed by appealing to its Lamperti-Kiu decomposition. We will finalise by illustrating the so-called Deep Wiener-Höpf factorisation of a related Markov Additive Process, in terms of the ascending and descending ladder potentials.

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--Dante Mata: "On the bailout dividend problem with periodic dividend payments and termination at an exponential time"

We study a version of the bailout optimal dividend problem driven by a spectrally negative Lévy process under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time, and with termination at an independent exponential time. We show the optimality of the Parisian-classical reflection strategy.

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--Zsofia Talyigas: "Genealogy and speed results on the N-particle branching random walk"

The $N$-particle branching random walk is a discrete time branching particle system with selection. We have $N$ particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size 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. In this talk I will first discuss the (very different) long-term behaviours of the speed of the particle cloud in cases of different jump distributions. Then, I will explain our main result, which concerns the genealogy of the population in the case when the jump distribution has polynomial tails and the population is large.

The result says that at a typical large time the genealogy of the population is given by a star-shaped coalescent.

This is joint work with Sarah Penington and Matthew Roberts.

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--Apolline Louvet: "A toy model for studying the consequences of the microscopic reproduction dynamics at the edge of an expanding population"

Studying the growth dynamics of stochastic growth models is a question of interest in population dynamics and percolation theory. Showing whether the growth is linear in time is generally already a challenging question, and there are few stochastic growth models for which the actual speed of growth is known. In this talk, I will show that the macroscopic expansion dynamics is heavily influenced by the microscopic reproduction dynamics at the front edge, which makes it challending to study the growth dynamics precisely. In order to do so, I will first use the example of a stochastic growth model for populations expanding in a continuum to show why the distinctive reproduction dynamics at the front edge makes intuition fail when trying to obtain the actual speed of growth. I will then introduce the 2-Columns Growth Process, which is a toy model for studying the effect of the reproduction dynamics at the front edge on the speed of growth. I will show how it is possible to obtain an explicit speed of growth for the process.

Based on a joint work with Amandine Véber (MAP5, Univ. Paris Cité).

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--Sam Jonhston: "Probability on trees and the Jacobian conjecture"

The Jacobian conjecture states that whenever $F:\mathbb{C}^n \to \mathbb{C}^n$ is a polynomial map (i.e. it has polynomial components), and the determinant of its Jacobian matrix is constant, then F is bijective and its inverse is a polynomial map.

It turns out that if $F:\mathbb{C}^n \to \mathbb{C}^n$ is any smooth function that is bijective in some neighbourhood, then the local inverse of F may be described in terms of multitype Galton-Watson trees with complex branching probabilities.

In this talk, we use the description of the inverse of a polynomial map in terms of multitype trees to study the Jacobian conjecture from a new combinatorial perspective.

Joint work with Elia Bisi, Piotr Dyszewski, Nina Gantert, Joscha Prochno, and Dominik Schmid.