The 9^th Workshop on Branching Processes and Related Topics

TianYuan Mathematics Research Center, Yunnan, China

2023/09/23 – 2023/09/28

Saturday 2023/09/23: Registration and Arrival

Thursday 2023/09/28: Departure

Titles and Abstracts

w   Bai, Tianyi (AMSS, Chinese Academy of Sciences)

Recent progresses in branching capacity

Abstract: Capacity is a fundamental concept with wide application in potential theory and random interlacements. It records information about hitting probabilities and Green's functions of a set. Replacing random walks by branching random walks in the definition of capacity, a parallel branching capacity is defined in [Zhu 2016].

Recently, there has been significant developments around this quantity in [Asselah, Schapira, Sousi 2023], [Schapira 2023] etc. In this talk, we give a walk-through for these works, and introduce our present result (joint work with Jean-Francois Delmas and Yueyun Hu) on the branching capacity of the range of a simple random walk and its continuous counterpart.

w   Chen, Xinxin (Beijing Normal University)

Conditioned Branching Random Walk

Abstract: TBA

w   He, Hui (Beijing Normal University)

Local limits for GW trees and Brownian trees

Abstract: We will review some recent results on local limits for GW trees and Brownian trees.

w   Hong, Jieliang (Southern University of Science and Technology)

On the critical probability of range-R bond percolation in six or less dimensions

Abstract: For the range-R bond percolation, the critical probability pc has been studied by Van der Hofstad and Sakai (2005) for d > 6 using lace expansion. In lower dimensions d \le 6, by connecting the bond percolation with the SIR epidemic model, we obtain the corresponding upper and lower bounds for the critical probability pc for R large, thus complementing the previous results in d > 6.

w   Hou, Haojie (Peking University)

Asymptotic expansion for branching killed Brownian motion with drift

Abstract: Let $Z_t^{(0,\infty)}$ be the point process formed by the positions of all particles alive at time $t$ in a branching Brownian motion with drift and killed upon reaching 0.

We study the asymptotic expansions of $Z_t^{(0,\infty)}(A)$ for $A= (a,b)$ and $A=(a,\infty)$ under the assumption that $\sum_{k=1}^\infty k(\log k)^{1+\lambda} p_k <\infty$ for large $\lambda$ in the regime of $\theta \in [0,\sqrt{2\beta(\mu-1})$. These results extend and sharpen the results of Louidor and Saglietti [J. Stat. Phys, 2020] and Kesten [Stochastic Process. Appl., 1978].

Jin, Peng (Beijing Normal University-Hong Kong Baptist University United International College)

Long-time behavior of affine processes

Abstract: Affine processes are Markov processes for which the logarithm of the characteristic function of its transition distribution is affine with respect to the initial state. This type of process unifies the concepts of continuous-state branching processes with immigration and Ornstein–Uhlenbeck type processes. In this talk I will first talk about recent results on the long-time behavior of traditional affine processes. In the second part I will talk about the recent extension of affine processes to the non-Markovian setting, the so-called Volterra affine processes. In particular, I will present our recent result on the existence of limiting distributions for the Volterra square-root process, a particular class of affine Volterra processes. This talk is based on joint works with Martin Friesen, Jonas Kremer and Barbara Rüdiger.

w   Ma, Heng (Peking University)

Extrema of two-type reducible branching Brownian motion

Abstract: Consider a two-type reducible branching Brownian motion in which particles' diffusion coefficients and branching rates are influenced by their types. Here reducible means that type $1$ particles can produce particles of type $1$ and type $2$, but type $2$ particles can only produce particles of type $2$. The maximum of this process is determined by two parameters: the ratio of the diffusion coefficients and the ratio of the branching rates for particles of different types. Belloum and Mallein [Electron. J. Probab. \textbf{26}(2021), no. 61] identified three phases of the maximum and the extremal process, corresponding to three regions in the parameter space.

We investigate how the extremal process behaves asymptotically when the parameters lie on the boundaries between these regions. An interesting consequence is that a double jump occurs in the maximum when the parameters cross the boundary of the so called anomalous spreading region. Furthermore if the parameters depend on time horizon $t$ and approach properly to the boundaries, the order of the maximum can interpolates smoothly between different regimes. Based on joint works with Yanxia Ren.

w   Peres, Yuval (BIMSA)

Controlled diffusion on Galton-Watson trees

Abstract: Given a unit of mass at the root of a Galton-Watson tree, how many mass splittings are needed to bring half the mass to distance n? In each split, the mass at a chosen vertex is divided equally among its neighbors. The surprising answer is related to random walk on these trees and to the maximum overhang problem. 

w   Shlosman, Senya (BIMSA)

Glassy trees

Abstract: I will talk about the Ising model on Cayley trees T_k. I will describe the difference between the Ising on trees T_k and on the regular lattices Z^d. In particular, on trees, one sees the glassy states. All the necessary definitions will be provided.

w   Sun, Zhenyao (Beijing Institute of Technology)

On the coming down from infinity of coalescing Brownian motions

Abstract: Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a necessary and sufficient condition for the number of particles to come down from infinity. We also identify the rate of this coming down from infinity for different initial configurations. This is a joint work with Clayton Barnes and Leonid Mytnik.

w   Véchambre, Grégoire (AMSS, Chinese Academy of Sciences)

Wright-Fisher diffusions in Lévy environments via combinatorics of some branching-coalescing processes

Abstract: We are interested in the evolution of a constant size population with two-types, modeled by a Wright-Fisher diffusion. We will present genealogy methods which are based on a branching-coalescing process called the Ancestral Selection Graph (ASG) that was introduced by Krone and Neuhauser in 1997. These methods allow to study interesting quantities like the probability of fixation of a type, the fixation time, or the distribution of types in the population after a long time. In recent works, we considered models where a Lévy environment drives the selection and, in particular, we allowed both types to have a selective advantage: depending on the random environment, selection sometimes favors an allele and sometimes the other. Because of the double selection, classical ASG methods fail in the case of this model. We propose a new combinatorics approach to study the ASG which allows, in the case of this model, to derive a series representation and a Taylor expansion for the fixation probability.

w   Xu, Wei (Beijing Institute of Technology)

Stochastic Volterra Equations for the Local Times of Spectrally Positive Stable Processes

Abstract: In this talk, we introduce the macroevolution mechanism of local times of a spectrally positive stable process in the spatial direction. Our main results state that conditioned on the finiteness of the first time at which the local time at zero exceeds a given value, the local times at positive half line are equal in distribution to the unique solution of a stochastic Volterra equation driven by a Poisson random measure whose intensity coincides with the Lévy measure. This helps us to provide not only a simple proof for the Hölder regularity, but also a uniform upper bound for all moments of the Hölder coefficient as well as a maximal inequality for the local times. Moreover, based on this stochastic Volterra equation, we extend the method of duality to establish an exponential-affine representation of the Laplace functional in terms of the unique solution of a nonlinear Volterra integral equation associated with the Laplace exponent of the stable process.

w   Zhou, Youzhou (Xi’an Jiaotong Liverpool University)

Transition Density of an Infinite-dimensional diffusion with the Jack Parameter

Abstract: From the Poisson-Dirichlet diffusions to the $Z$-measure diffusions, they all have explicit transition densities. In this paper, we will show that the transition densities of the $Z$-measure diffusions can also be expressed as a mixture of a sequence of probability measures on the Thoma simplex. The coefficients are still the transition probabilities of the Kingman coalescent stopped at state $1$. This fact will be uncovered by a dual process method in a special case where the $Z$-measure diffusions is established through up-down chain in the Young graph.

参会人员名单