Franco Arrejoría, Pablo Groisman, Leonardo T. Rolla
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.
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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 $X$.
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.
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.
In this paper we describe the subcritical contact process on $\Z^d$ for large times, starting with all sites infected. The configuration is described in terms of the macroscopic locations of infected regions in space and the relative positions of infected sites in each such region.
In this paper we are concerned with threshold-one contact processes on lattices. We show that the probability that the origin is infected converges to 0 at an exponential rate I in the subcritical case. Furthermore, we give a limit theorem for I as the degree of the lattice grows to infinity. Our results also hold for classic contact processes on lattices.
We consider the subcritical contact branching random walk on Zd in continuous time with the arbitrary number of offspring and with immigration. We prove the existence of the steady state (statistical equilibrium).
We study a non-equilibrium dynamical model: a marked continuous contact model in $d$-dimensional space, $d \ge 1$. In contrast with the continuous contact model in a critical regime, see \cite{KKP}, \cite{KPZ}, the model under consideration is in the subcritical regime and it contains an additional spontaneous spatially homogeneous birth from an external source. We prove that this system has an invariant measure. We prove also that the process starting from any initial distribution converges to this invariant measure.
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.
The Contact Process Can Survive on a Slightly Subcritical Dynamical Percolation Cluster
该研究证明在略微亚临界动态渗透簇上，接触过程可以生存，扩展了一维情况下的结果。
This paper gives a new, simple proof of the known fact that for contact processes on general lattices, in the subcritical regime the expected number of infected sites decays exponentially fast as time tends to infinity. The proof also yields an explicit bound on the survival probability below the critical recovery rate, which shows that the critical exponent associated with this function is bounded from below by its mean-field value. The main idea of the proof is that if the expected number of infected sites decays slower than exponentially, then this implies the existence of a harmonic functi…
What is the long-time behavior of the law of a contact process started with a single infected site, distributed according to counting measure on the lattice? This question is related to the configuration as seen from a typical infected site and gives rise to the definition of so-called eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to a multiplicative constant. In this paper, we study eigenmeasures of contact processes on general countable groups in the subcritical regime. We prove that in this regime, the proc…
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 contact process on dynamic regular graphs: monotonicity and subcritical phase
本文研究动态随机 $d$-正则图上的接触过程及 herds 过程，证明临界感染率随边动态参数严格递减、亚临界灭绝时间呈指数尾，并推出全感染初始态下灭绝时间对顶点数为对数阶。
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.
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。
In this paper we are concerned with the contact process with random recovery rates and edge weights on complete graph with $n$ vertices. We show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate. In the subcritical case, the process dies out before a moment with order $O(\log n)$ with high probability as $n\rightarrow+\infty$. In the supercritical case, the process survives at a moment with order $\exp\{O(n)\}$ with high probability as $n\rightarrow+\infty$. Our proof for the subcri…
We study an iterated temporal and contemporaneous aggregation of $N$ independent copies of a strongly stationary subcritical Galton-Watson branching process with regularly varying immigration having index $α\in (0, 2)$. Limits of finite dimensional distributions of appropriately centered and scaled aggregated partial sum processes are shown to exist when first taking the limit as $N \to \infty$ and then the time scale $n \to \infty$. The limit process is an $α$-stable process if $α\in (0, 1) \cup (1, 2)$, and a deterministic line with slope $1$ if $α= 1$.
Subcritical epidemics on random graphs
本文研究随机图上低感染率接触过程的生存时间，证明在 Erdős-Rényi 随机图中高密度局部构型使生存时间达到 n^{λ^{2+o(1)}}，并通过修正接触过程建立匹配上界，方法可推广至具有指数矩的一般度分布随机图。
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.
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。
We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. This generalizes and simplifies an earlier result of Van den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase transition on $\mathbb{Z}^2$. Our proof relies on the OSSS inequality for Boolean functions and is inspired by a series of papers by Duminil-Copin, Raou…
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。
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。
We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cramér condition the stationary distribution has a power law tail. We determine the tail process of the stationary Markov chain, prove point process convergence, and convergence of the partial sums. The original motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which connects a random walk in a random environment model to a special Galton-Watson process with immigration in a random environment. We obtain new …
We study the stationary distribution of the (spread-out) $d$-dimensional contact process from the point of view of site percolation. In this process, vertices of $\mathbb{Z}^d$ can be healthy (state 0) or infected (state 1). With rate one infected sites recover, and with rate $λ$ they transmit the infection to some other vertex chosen uniformly within a ball of radius $R$. The classical phase transition result for this process states that there is a critical value $λ_c(R)$ such that the process has a non-trivial stationary distribution if and only if $λ> λ_c(R)$. In configurations sampled from…
The stacked contact process is a three-state spin system that describes the co-evolution of a population of hosts together with their symbionts. In a nutshell, the hosts evolve according to a contact process while the symbionts evolve according to a contact process on the dynamic subset of the lattice occupied by the host population, indicating that the symbiont can only live within a host. This paper is concerned with a generalization of this system in which the symbionts may affect the fitness of the hosts by either decreasing (pathogen) or increasing (mutualist) their birth rate. Standard c…
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 investigate the contact process on four different types of scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where each potential edge is updated with a rate depending on the strength of the adjacent vertices. Depending on the type of graph, the tail exponent of the degree distribution and the updating rate, we find parameter regimes of fast and slow extinction and in the latter case identify metastable exponents that undergo first order phase transitions.
The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if H_t denotes the set of already occupied sites at time t, we show that for almost every environment, when the contact process survives, the set H_t/t almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem.
Stationary solution to Stochastically Forced Euler-Poisson Equations in Bounded Domain: Part 2. 1-D Ohmic Contact Boundary
本文针对带欧姆接触边界的一维随机欧拉-泊松方程，利用Banach不动点定理和加权能量估计证明稳态附近解的全局存在性与渐近稳定性，并证明不变测度为稳态对应的Dirac测度。
The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if H_t denotes the set of already occupied sites at time t, we show that for almost every environment, when the contact process survives, the set H_t/t almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem.
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。
We consider a chain composed of $N$ coupled harmonic oscillators in contact with heat baths at temperature $T_\ell$ and $T_r$ at sites 1 and $N$ respectively. The oscillators are also subjected to non-momentum conserving bulk stochastic noises. These make the heat conductivity satisfy Fourier's law. Here we describe some new results about the hydrodynamical equations for typical macroscopic energy and displacement profiles, as well as their fluctuations and large deviations, in two simple models of this type.
We consider the zero-range process with long jumps and in contact with infinitely many reservoirs in its non-equilibrium stationary state. We derive the hydrostatic limit and the Fick's law, which are a consequence of a relationship between the exclusion process and the zero-range process. We also obtain the large deviation principle for the empirical density, i.e. we compute the non-equilibrium free energy.
We investigate the macroscopic time evolution and stationary states of a mean field generalized contact process in $\mathbb{R}^d$. The model is described by a coupled set of nonlinear integral-differential equations. It was inspired by a model of neurons with discrete voltages evolving by a stochastic integrate and fire mechanism. We obtain a complete solution in the spatially uniform case and partial solutions in the general case. The system has one or more fixed points and also traveling wave solutions.
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