会议日程

 

 

报告安排

 

题目与摘要

 

CSBPs, Ancestral Lineages and Siegmund duality

Clément Foucart (法国巴黎北索邦大学)

2022年8月23日 16:00-18:00

摘要：In the first session, I will recall briefly some well-known facts on CSBPs, as for  instance: the representation of their semigroup, of the generator and the classification of the boundaries. I will refer to Zenghu Li's lecture notes and Andreas Kyprianou's book for those results and do not provide proofs. I shall introduce some of the basic notations and explain heuristically what we are going to study. Our starting point will be a construction due to Bertoin and Le Gall of a complete population model (i.e. where individuals are specified) for CSBPs (existence will be omitted, I refer to Bertoin's lecture notes). We shall end the session by defining a new process that represents the ancestral lineage of individuals backwards in time. This corresponds to the inverse flow of the CSBPs, we call them Ancestral Lineage Processes (ALPs), and provide some first information on their semigroup, longterm behaviors and generators. Depending on the time left, I shall give details on the proof leading to the generator form of ALPs.

 

Genealogy and Coalescent in CSBPs

Clément Foucart (法国巴黎北索邦大学)

2022年8月25日 16:00-18:00

摘要：The objective of this session is to explain how ancestral lineages merge as time goes backwards. We shall see how to sample individuals in the current generation along an independent Poisson process, and how lineages of those individuals merge.  The coalescent theory induced behind is rather elementary and shares many features with flows of bridges and exchangeable coalescents. I plan to provide some details on how the coalescent process evolves in time.

The long term behavior of the partition-valued coalescent will be discussed; in the subcritical case, we shall see that it converges almost surely towards a partition with infinitely many blocks each independent, whose size law is related to the quasi-stationary distribution. Last, we will state some results on the long-term behavior of the ALP processes in the subcritical case and find an almost sure renormalisation of it. 

 

Logistic CSBPs, Laplace duality and reflection at infinity.

Clément Foucart (法国巴黎北索邦大学)

2022年8月26日 16:00-18:00

摘要：In the last session, we change of topic and introduce a new force in the CSBPs; the so-called logistic competition (or quadratic competition). It is modelling a phenomenon of pairwise fight between individuals in the population (in terms of population modelling, this can be viewed as modelling limited ressources). The process with competition (LCSBP) does not satisfy the branching property; but we shall be able to study it through another duality relationship with a certain one-dimensional diffusion on $(0,\infty)$: the so-called Laplace duality. We shall classify the behaviors of the process at its boundary infinity with the help of the dual. A phenomenon of instantaneous reflection at infinity is revealed. The construction we give of LCSBP reflected at infinity is made by taking limits of processes with boundary entrance; I will spend some time on this construction. The latter is  not providing any information on the excursion measure, I shall state some results on the local time at infinity of the LCSBP by introducing the Siegmund dual process of the Laplace dual.

 

Local limits theory and pruning processes for Galton-Watson trees.

何辉 (北京师范大学)

2022年8月23日 10:00-12:00

摘要：First, we will give a general presentation on local limit of Galton-Watson (BGW) trees conditioned to be large, where the limit is again a GW tree with a distinguished spine which is either infinite or finite. Second, some pruning procedures on trees will be introduced and applied to Galton-Watson trees. In this way, some tree-valued Markov processes and related cut trees are induced.

 

From Galton-Watson trees to Levy trees: scaling limits and characterizations.

何辉 (北京师范大学)

2022年8月25日 14:00-16:00

摘要：We show that Galton-Watson trees, suitably scaled, converge to a kind of continuum random trees, so-called Levy trees, whose characterizations will be also given. Then the tree-valued processes and cut trees introduced in the first part, suitably scaled, will be shown to converge to their continuum counterparts.

 

Some applications of Levy trees to study branching processes.

何辉 (北京师范大学)

2022年8月26日 10:00-12:00

摘要：Levy trees give the genealogy of continuous state branching processes. We shall use Levy trees to study the continuous state branching processes, including local limit and population structures. We will focus on quadratic case.

 

An introduction to superprocess: as the scaling limit of branching particle system

侯浩杰（北京大学）

2022年8月23日 14:00-16:00

摘要：Firstly we introduce the integral equation for branching Brownian motion(BBM), under some condition, we can get the scaling limit for a family of BBMs to super-Brownian motion. Also, similar argument also holds for general branching Markov process to superprocess. Then we introduce some properties and tools in superprocess.

 

Spine decomposition for super-Brownian motion.

侯浩杰（北京大学）

2022年8月24日 10:00-12:00

摘要：In this talk, we introduce three martingales--additive martingale, truncated martingale and derivative martingale. Then two class of spine decomposition are discussed. Although the idea in the proof of the convergence for these martingales is quite similar to those in BBM or branching random walk(BRW), we may talk about some difficulties in super-Brownian motion.

 

Skeleton decomposition for superprocess.

侯浩杰（北京大学）

2022年8月25日 10:00-12:00

摘要：In this part, we talk about the skeleton decomposition for superprocess. For applications, we use some known results from BBM or BRW to  prove some results in super-Brownian motion.

 

 

Competition and condensation in some population models

袁玲珑（英国利物浦大学）

2022年8月24日 14:00-16:00

摘要：In the first talk, we will start with Kingman’s model on the competition between selection and mutation (1978).  Kingman proved the convergence of the fitness distributions and described in particular the phase transition for the occurrence of condensation on the largest fitness value. More precisely, the condensation occurs if selection dominates mutation, and vice versa.  Bianconi and Barabási (2000) argued that this phenomenon can be mapped to the Bose-Einstein condensation and used a preferential attachment model with fitness to illustrate this mapping. Indeed, there are many generalisations of Kingman’s model introduced recently, that retain the feature of a pair of competing forces and the phase transition of condensation occurrence. These generalisations include branching processes, preferential attachment models with fitness and random permutation models. I will present some of these models and show that some universal characteristics exist for the condensation. 

 

Kingman’s model with random mutation probabilities

袁玲珑（英国利物浦大学）

2022年8月26日 14:00-16:00

摘要：Kingman’s model is a haploid population model of discrete time, infinite population size and fixed mutation probability for all generations. In the second talk, I will introduce Kingman’s model with i.i.d. mutation probabilities and analyse its convergence and condensation. It is one example of Kingman’s model in a random environment. The particular emphasis will be on the comparison between the original model and random model. Of particular interest is the question whether the extra randomness will enhance or weaken the condensation. A universal conjecture is that the effects will be the same across many Kingman-like models. A finer analysis of the random Kingman’s model for the future work will need insights from renewal sequences and random matrices.