Benedikt Stufler
We prove local convergence results of rerooted conditioned multi-type Galton--Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine.
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We demonstrate a novel strong law of large numbers for branching processes, with a simple proof via measure-theoretic manipulations and spine theory. Roughly speaking, any sequence of events that eventually occurs almost surely for the spine entails the almost sure convergence of a certain sum over particles in the population.
In the literature, the spine decomposition of branching Markov processes was constructed under the assumption that each individual has at least one child. In this paper, we give a detailed construction of the spine decomposition of general branching Markov processes allowing the possibility of no offspring when a particle dies. Then we give some applications of the spine decomposition.
In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behaviour is then deduced from renewal theorems and induction.
Using the foundations laid down in Hardy and Harris (2006) ["A new formulation of the spine approach in branching diffusions", arXiv:math.PR/0611054], we present new spine proofs of the L^p-convergence p>=1) of some key `additive' martingales for three distinct models of branching diffusions, including new results for a multi-type branching Brownian motion and discussion of left-most particle speeds. The spine techniques we develop give clear and simple arguments in the spirit of the conceptual spine proofs found in Kyprianou (2004) ["Travelling wave solutions to the KPP equation, Ann. Inst. H…
In this paper, we provide a pathwise spine decomposition for superprocesses with both local and non-local branching mechanisms under a martingale change of measure. This result complements the related results obtained in Evans (1993), Kyprianou et al. (2012) and Liu, Ren and Song (2009) for superprocesses with purely local branching mechanisms and in Chen, Ren and Song (2016) and Kyprianou and Palau (2016) for multitype superprocesses. As an application of this decomposition, we obtain necessary/sufficient conditions for the limit of the fundamental martingale to be non-degenerate. In particul…
We consider a branching-selection particle system on the real line, introduced by Brunet and Derrida. In this model the size of the population is fixed to a constant $N$. At each step individuals in the population reproduce independently, making children around their current position. Only the $N$ rightmost children survive to reproduce at the next step. Bérard and Gouéré studied the speed at which the cloud of individuals drifts, assuming the tails of the displacement decays at exponential rate; Bérard and Maillard took interest in the case of heavy tail displacements. We take interest in an …
We present a formalization of the spine change of measure approach for branching diffusions that improves on the scheme laid out for branching Brownian motion in Kyprianou (2004) ["Travelling wave solutions to the KPP equation, Ann. Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72] which itself made use of earlier works of Lyons et al (1997) ["A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes", Classical and modern branching processes, IMA Vol. Math. Appl., vol.84, Springer, New York, pp181-185]. We use our new formulation to interpret certain `Gibbs-Boltzm…
We consider branching random walks with a spine in the domain of attraction of an $α$-stable Lévy process. For this process, the classical derivative martingale in general degenerates in the limit. We first determine the quantity replacing the derivative martingale and show that it converges to a non-degenerate limit under a certain LlogL-type condition which we assume to be optimal. We go on to give the Seneta-Heyde norming for the critical additive martingale under the same assumptions. The proofs are based on the methods introduced in our previous paper which considered the finite variance …
We show uniqueness of the spine of a Fleming-Viot particle system under minimal assumptions on the driving process. If the driving process is a continuous time Markov process on a finite space, we show that asymptotically, when the number of particles goes to infinity, the branching rate for the spine is twice that of a generic particle in the system, and every side branch has the distribution of the unconditioned generic branching tree.
Evans (1992) described the semi-group of a superprocess with quadratic branching mechanism under a martingale change of measure in terms of the semi-group of an immortal particle and the semigroup of the superprocess prior to the change of measure. This result, commonly referred to as the spine decomposition, alludes to a pathwise decomposition in which independent copies of the original process `immigrate' along the path of the immortal particle. For branching particle diffusions the analogue of this decomposition has already been demonstrated in the pathwise sense, see for example Hardy and …
Spectral analysis and $k$-spine decomposition of inhomogeneous branching Brownian motions. Genealogies in fully pushed fronts
本文研究完全推动型波动前沿中非均匀分支布朗运动的谱系结构，通过建立Yaglom律、脊分解和谱分析方法，证明了粒子谱系收敛于Brownian Coalescent Point Process，并确定了脊的不变测度与混合时间。
Consider a supercritical superdiffusion (X_t) on a domain D subset R^d with branching mechanism -β(x) z+α(x) z^2 + int_{(0,infty)} (e^{-yz}-1+yz) Pi(x,dy). The skeleton decomposition provides a pathwise description of the process in terms of immigration along a branching particle diffusion. We use this decomposition to derive the Strong Law of Large Numbers (SLLN) for a wide class of superdiffusions from the corresponding result for branching particle diffusions. That is, we show that for suitable test functions f and starting measures mu, < f,X_t>/P_{mu}[< f,X_t>] -> W_{infty}, P_{mu}-almost …
In this paper we use the spine decomposition and martingale change of measure to establish a Kesten-Stigum $L\log L$ theorem for branching Hunt processes. This result is a generalization of the results in Asmussen-Hering (1976) and Hering (1978) for branching diffusions.
We study a queueing system with Poisson arrivals on a bus line indexed by integers. The buses move at constant speed to the right and the time of service per customer getting on the bus is fixed. The customers arriving at station i wait for a bus if this latter is less than d\_i stations before, where d\_i is non-decreasing. We determine the asymptotic behavior of a single bus and when two buses eventually coalesce almost surely by coupling arguments. Three regimes appear, two of which leading to a.s. coalescing of the buses.The approach relies on a connection with aged structured branching pr…
In this paper we prove that, under certain conditions, a strong law of large number holds for a class of branching particle systems $X$ corresponding to the parameters $(Y,β,ψ)$, where $Y$ is a Hunt process and $ψ$ is the generating function for the offspring. The main tool of this paper is the spine decomposition and we only need a $L\log L$ condition.
In this paper, we provide a pathwise spine decomposition for multitype superdiffusions with non-local branching mechanisms under a martingale change of measure. As an application of this decomposition, we obtain a necessary and sufficient condition (called the $L\log L$ criterion) for the limit of the fundamental martingale to be non-degenerate. This result complements the related results for superprocesses with purely local branching mechanisms and in Kyprianou and Palau (2016) for super Markov chains.
We study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating travelling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by representing the solutions probabilistically in terms of the limits of the additive and derivative martingales. We also prove that there is no pulsating travelling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.
Spinal constructions for continuous type-space branching processes with interactions
针对连续时间下结构化相互作用种群的分支过程问题，提出基于脊构造的Girsanov型结果与多对一公式，推导含相互作用的广义连续时间Kesten-Stigum定理，还给出用于规模依赖随机种群精确模拟的脊构造替代方法。
From multitype branching Brownian motions to branching Markov additive processes
本文提出了一种多型分支莱维过程的框架，利用脊分解技术证明了加性鞅的收敛性和旅行波解的存在性，解决了on-off分支布朗运动模型中的若干开放问题。
We define a doubly infinite, monotone labeling of Bienayme-Galton-Watson (BGW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i; i\ge 1)$, where $A_i$ is the coalescence time between individuals i and i+1. There is a Markov process of point measures $(B_i; i\ge 1)$ keeping track of more ancestral relationships, such that $A_i$ is also the first point mass of $B_i$. This process of point measures is also closely related to an inhomogeneous spine decomposition of the lineage of the first surviving particle in gene…
Let $η^*_n$ denote the maximum, at time $n$, of a nonlattice one-dimensional branching random walk $η_n$ possessing (enough) exponential moments. In a seminal paper, Aidekon demonstrated convergence of $η^*_n$ in law, after recentering, and gave a representation of the limit. We give here a shorter proof of this convergence by employing reasoning motivated by Bramson, Ding and Zeitouni. Instead of spine methods and a careful analysis of the renewal measure for killed random walks, our approach employs a modified version of the second moment method that may be of independent interest.
Intersections of branching random walks on $\mathbb{Z}^8$
本文研究$\u2124^8$上以无穷不变树为索引的分支随机游走的相交问题，得到两类不同索引的独立游走间不相交概率的精确阶数，并证明分支随机游走值域的分支容量满足弱大数定律。
Minimum and extremal process for a branching random walk outside the boundary case
本文研究了超临界分支随机游走在边界情况外的最小和极值过程，验证了相关猜想并得出了弱收敛性和下确界结果。
We consider the Brownian tree introduced by Aldous and the associated Q-process which consists in an infinite spine on which are grafted independent Brownian trees. We present a reversal procedure on these trees that consists in looking at the tree downward from its top: the branching points becoming leaves and leaves becoming branching points. We prove that the distribution of the tree is invariant under this reversal procedure, which provides a better understanding of previous results from Bi and Delmas (2016).
We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate theta, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have i.i.d. lifetimes durations (non necessarily exponential) during which they give birth independently at constant rate b. First, using spine decomposition, we relax previously known assumptions required for a.s. convergence of total population size. Then, we consider three models of structured populations: …
We define generic ensembles of infinite trees. These are limits as $N\to\infty$ of ensembles of finite trees of fixed size $N$, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension $d_h =2$. Our main result is that their spectral dimension is $d_s=4/3$, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is 1/3.
The all-time maximum for branching Brownian motion with absorption conditioned on long-time survival
针对带吸收的分支布朗运动最终以正概率灭绝的特性，研究者以过程长时间存活为条件，利用脊分解分析其小时间行为，证明粒子距原点最远的时间阶为$t^{5/6}$。
We first study a model, introduced recently in \cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no `obstacle' placed there. The obstacles appear at each site with probability $p\in [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $\mathbb P_p$-measure, as $n\to\infty$, (i) Critical case: P^ω(S…
We study the high temperature phase of a family of typed branching diffusions initially studied in [Astérisque 236 (1996) 133--154] and [Lecture Notes in Math. 1729 (2000) 239--256 Springer, Berlin]. The primary aim is to establish some almost-sure limit results for the long-term behavior of this particle system, namely the speed at which the population of particles colonizes both space and type dimensions, as well as the rate at which the population grows within this asymptotic shape. Our approach will include identification of an explicit two-phase mechanism by which particles can build up i…
Convergence of genealogies through spinal decomposition with an application to population genetics
本文提出一种基于矩方法和k-spine分解的一般框架，证明分支马尔可夫过程谱系在Gromov-弱拓扑下的收敛性，并将其应用于具有重组的双亲Wright-Fisher模型，揭示其极限行为兼具临界与超临界特征。
Uniform sampling of multitype continuous-time Bienaymé-Galton-Watson trees
研究连续时间多类型Bienaymé-Galton-Watson过程在固定时间点均匀无放回抽取k个个体的谱系，通过k-脊分解和测度变换，揭示类型间复杂交互，为后续研究奠定基础。
We study a special case of the vertex splitting model which is a recent model of randomly growing trees. For any finite maximum vertex degree $D$, we find a one parameter model, with parameter $α\in [0,1]$ which has a so--called Markov branching property. When $D=\infty$ we find a two parameter model with an additional parameter $γ\in [0,1]$ which also has this feature. In the case $D = 3$, the model bears resemblance to Ford's $α$--model of phylogenetic trees and when $D=\infty$ it is similar to its generalization, the $αγ$--model. For $α= 0$, the model reduces to the well known model of pref…
Behaviour near explosion in continuous-time Galton-Watson trees
本研究针对连续时间Galton-Watson树在有限时间爆炸的现象，定义了爆炸脊柱，证明脊柱外出生呈泊松描述，爆炸前人口近似伽马分布，并证明重标度过程收敛于带移民的次临界连续状态分支过程，其时间反转具有马尔可夫性，关联合并过程为随机时间变换的Beta-合并过程。
Let $τ$n be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n $\in$ N *) is a deterministic positive sequence. We study the local limit of these trees $τ$n as n $\rightarrow$ $\infty$ and observe three distinct regimes: if (an, n $\in$ N *) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an…
The Genealogy of Conditioned Critical Galton--Watson Trees
本文研究临界Galton--Watson过程的条件树的左右子树谱系，证明其溯祖时间序列的收敛性与渐近独立性，并以此概率证明临界GW过程的条件极限定理。
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