Rongli Liu, Yan-Xia Ren, Renming Song, Zhenyao Sun
Suppose that $X$ is a subcritical superprocess. Under some asymptotic conditions on the mean semigroup of $X$, we prove the Yaglom limit of $X$ exists and identify all quasi-stationary distributions of $X$.
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We consider a class of subcritical superprocesses $(X_t)_{t\geq 0}$ with general spatial motions and general branching mechanisms. We study the asymptotic behaviors of $\mathbf Q_{t,r}$, the distribution of $X_t$ conditioned on $X_{t+r}$ not being a null measure. We first give the existence of $\lim_{t\to \infty}\mathbf Q_{t,r}$ and $\lim_{r\to \infty}\mathbf Q_{t,r}$, and then show that an $L\log L$-type condition is equivalent to the existence of the double limits: $\lim_{r\to \infty} \lim_{t\to\infty}\mathbf Q_{t, r}$ and $\lim_{t\to \infty} \lim_{r\to\infty}\mathbf Q_{t, r}$. Finally, when…
It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritial superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is individuals who produce eternal genealogical lines of decent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess…
Let $ξ$ be a Dawson--Watanabe superprocess in $\mathbb{R}^d$ such that $ξ_t$ is a.s. locally finite for every $t\geq 0$. Then for $d\geq2$ and fixed $t>0$, the singular random measure $ξ_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoods of $\operatorname {supp}ξ_t$. When $d\geq3$, the local distributions of $ξ_t$ near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure $\tildeξ$. By contrast, the corresponding distributions for $d=2$ are locally invariant. Fur…
In this short note we obtain necessary and sufficient conditions for the existence of the moments of the stationary distribution of a subcritical multitype Galton-Watson process with immigration.
We show that the quasi-stationary distribution of the subcritical contact process on $\mathbb{Z}^d$ is unique. This is in contrast with other processes which also do not come down from infinity, like stable queues and Galton-Watson, and it seems to be the first such example.
Consider a locally finite Dawson-Watanabe superprocess $ξ=(ξ_t)$ in $\mathsf{R}^d$ with $d\geq2$. Our main results include some recursive formulas for the moment measures of $ξ$, with connections to the uniform Brownian tree, a Brownian snake representation of Palm measures, continuity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of $ξ_t$ by a stationary cluster $\tildeη$ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditiona…
We show the existance of the stationary distributions of subcritical multitype Galton-Watson processes without any conditions on the mean matrix of the offspring distributions. Some additional properties of the stationary distribution are also proven.
Quasi-stationary distributions for subcritical branching Markov chains
本文研究了亚临界分支马尔可夫链，基于 spinal decomposition 和 many-to-few formula，通过矩方法给出了 Yaglom 极限存在性的概率证明，并提供了所有拟平稳分布的显式积分表示，无需依赖 Martin 边界理论。
The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
A $λ$-invariant measure of a sub-Markov chain is a left eigenvector of its transition matrix of eigenvalue $λ$. In this article, we give an explicit integral representation of the $λ$-invariant measures of subcritical Bienaymé--Galton--Watson processes killed upon extinction, i.e.\ upon hitting the origin. In particular, this characterizes all quasi-stationary distributions of these processes. Our formula extends the Kesten--Spitzer formula for the (1-)invariant measures of such a process and can be interpreted as the identification of its minimal $λ$-Martin entrance boundary for all $λ$. In t…
The Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift
本文研究带吸收的次临界分支布朗运动，证明存活条件下过程收敛于 Yaglom 拟平稳分布，并分析漂移趋近临界值时的粒子数、最高位置及经验分布渐近行为。
We describe the asymptotic behavior of the conditional least squares estimator of the offspring mean for subcritical strongly stationary Galton--Watson processes with regularly varying immigration with tail index $α\in (1,2)$. The limit law is the ratio of two dependent stable random variables with indices $α/2$ and $2α/3$, respectively, and it has a continuously differentiable density function. We use point process technique in the proofs.
In the first part of this paper we consider a general stationary subcritical cluster model in $\mathbb{R}^d$. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein-Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE-solution in the special case of a Poisson driven random connection model.
Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.
Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). Individuals reproduce independently according to the same subcritical offspring distribution. We study the long-term behaviour of the ancestral lineages as time goes to the past and show that the flow of ancestral lineages, properly renormalized, converges almost surely to the inverse of a drift-free subordinator whose Laplace exponent is explicit in terms of the branching mechanism. We provide an interpretation in terms of the genealogy of the population. In particular, we s…
Quasi-limiting behaviour of the sub-critical Multitype Bisexual Galton-Watson Branching Process
本文研究次临界双性Galton-Watson分支过程的拟极限行为，利用凹算子特征值与灭绝的联系，克服缺乏分支性质的困难，证明拟平稳分布的存在性与收敛性。
Domains of Attraction for Quasi-Stationary Distributions of Branching Processes in Random Environments
本文完整刻画了布朗随机环境中次临界稳定连续状态分支过程的拟平稳分布及其吸引域，给出了收敛的充要条件。
Existence of subcritical percolation phases for generalised weight-dependent random connection models
本文研究带标记泊松点过程上的广义权重依赖随机连接模型，通过控制长边给出临界环带穿越强度为正及次临界相存在的充分条件，并建立典型簇直径与规模的尾界。
We consider the connected component of the partial duplication model for a random graph, a model which was introduced by Bhan, Galas and Dewey as a model for gene expression networks. The most rigorous results are due to Hermann and Pfaffelhuber, who show a phase transition between a subcritical case where in the limit almost all vertices are isolated and a supercritical case where the proportion of the vertices which are connected is bounded away from zero. We study the connected component in the subcritical case, and show that, when the duplication parameter $p<e^{-1}$, the degree distributi…
On large queue lengths in generalised Jackson networks
本文研究了一般化Jackson网络中队列长度的大偏差原理，通过Cramer条件推导出准势能函数。
Motivated by the study of the metastable stochastic Ising model at subcritical temperature and in the limit of a vanishing magnetic field, we extend the notion of ($κ$, $λ$)-capacities between sets, as well as the associated notion of soft-measures, to the case of overlapping sets. We recover their essential properties, sometimes in a stronger form or in a simpler way, relying on weaker hypotheses. These properties allow to write the main quantities associated with reversible metastable dynamics, e.g. asymptotic transition and relaxation times, in terms of objects that are associated with two-…
We study the long-time behavior of affine processes on positive self-adjoiont Hilbert-Schmidt operators which are of pure-jump type, conservative and have finite second moment. For subcritical processes we prove the existence of a unique limit distribution and construct the corresponding stationary affine process. Moreover, we obtain an explicit convergence rate of the underlying transition kernels to the limit distribution in the Wasserstein distance of order $p\in [1, 2]$ and provide explicit formulas for the first two moments of the limit distribution. We apply our results to the study of i…
On the uniqueness of quasi-stationary distributions for population models with spatial structure
本文研究了具有空间结构的人口模型中准稳分布的唯一性，通过引入谱系和分支随机游走，证明了其在添加少量额外信息后具有唯一性。
Stationary measures and the continuous-state branching process conditioned on extinction
本文研究了几乎必然灭绝的连续状态分支过程，通过Lévy过程尺度函数表示平稳测度，并证明了在临界与次临界情形下条件灭绝过程的极限定理及非退化极限分布。
The stationary asymptotic properties of the diffusion limit of a multi-type branching process with neutral mutations are studied. For the critical and subcritical processes the interesting limits are those of quasi-stationary distributions conditioned on non-extinction. Pedagogical derivations are given for known results that the limiting distributions for supercritical and critical processes are found to collapse onto rays aligned with stationary eigenvectors of the mutation rate matrix, in agreement with discrete multi-type branching processes. For the sub-critical process the previously uns…
We provide a generalization of Theorem 1 in Bartkiewicz, Jakubowski, Mikosch and Wintenberger (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly sta…
We study a stationary Gibbs particle process with deterministically bounded particles on Euclidean space defined in terms of an activity parameter and non-negative interaction potentials of finite range. Using disagreement percolation we prove exponential decay of the correlation functions, provided a dominating Boolean model is subcritical. We also prove this property for the weighted moments of a U-statistic of the process. Under the assumption of a suitable lower bound on the variance, this implies a central limit theorem for such U-statistics of the Gibbs particle process. A byproduct of o…
Minimum stationary values of sparse random directed graphs
针对有界度有向配置模型上的简单随机游走，在最小出度≥2时证明唯一平稳分布存在，确定最小正平稳值为$n^{-(1+C+o(1))}$，并揭示C由入邻域和轨迹两因素决定，补充了已有入度相关研究。
The stochastic Kuramoto model defined on a sequence of graphs is analyzed: the emphasis is posed on the relationship between the mean field limit, the connectivity of the underlying graph and the long time behavior. We give an explicit deterministic condition on the sequence of graphs such that, for any finite time and any initial condition, even dependent on the network, the empirical measure of the system stays close to the solution of the McKean-Vlasov equation associated to the classical mean field limit. Under this condition, we study the long time behavior in the subcritical and in the s…
Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise
该研究针对受时空白噪声导数扰动的奇异表面准地转方程，通过改进的Da Prato--Debussche技巧与凸积分迭代，首次证明了次临界、临界及超临界情形下无穷多非高斯概率强解和遍历平稳解的存在性。
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