State-dependent Foster-Lyapunov criteria for subgeometric convergence of Markov chains

具有确定性等待时间阈值的基于技能队列的稳定性

Stability of skill-based queues with deterministic waiting time thresholds

研究具有确定性等待时间阈值的基于技能队列的稳定性，通过将排队等待时间过程表述为PDMP并构建李雅普诺夫函数，建立了该模型的充要稳定性条件，证明阈值不影响稳定性。

Fecundity regulation in a spatial birth-and-death process

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster-Lyapunov-type function, we study return times to certain level sets of tempered configurations. We find also sufficient conditions that the degenerate invariant distribution is unique for the considered process.

有界Lipschitz范数下指数遍历马尔可夫过程的中心极限定理

The central limit theorem for Markov processes that are exponentially ergodic in the bounded-Lipschitz norm

本文针对满足有界Lipschitz距离指数遍历与连续Foster-Lyapunov条件的Markov-Feller连续时间过程建立中心极限定理，并以一类Poisson跳跃切换的分段确定性马尔可夫过程为例验证了假设成立。

Geometric Ergodicity of the multivariate COGARCH(1,1) Process

For the multivariate COGARCH(1,1) volatility process we show sufficient conditions for the existence of a unique stationary distribution, for the geometric ergodicity and for the finiteness of moments of the stationary distribution by a Foster-Lyapunov drift condition approach. The test functions used are naturally related to the geometry of the cone of positive semi-definite matrices and the drift condition is shown to be satisfied if the drift term of the defining stochastic differential equation is sufficiently `negative'. We show easily applicable sufficient conditions for the needed irred…

Asymptotically Tight Steady-State Queue Length Bounds Implied By Drift Conditions

The Foster-Lyapunov theorem and its variants serve as the primary tools for studying the stability of queueing systems. In addition, it is well known that setting the drift of the Lyapunov function equal to zero in steady-state provides bounds on the expected queue lengths. However, such bounds are often very loose due to the fact that they fail to capture resource pooling effects. The main contribution of this paper is to show that the approach of "setting the drift of a Lyapunov function equal to zero" can be used to obtain bounds on the steady-state queue lengths which are tight in the heav…

时间非齐次 SDE 半群的入口测度：可能退化且扩张的情形

Entrance measures for semigroups of time-inhomogeneous SDEs: possibly degenerate and expanding

本文推广 Harris 小集方法，建立时间非齐次、可退化且扩张的 SDE 向入口测度的次几何收敛，并证明拟周期测度及环面不变测度的存在唯一性与遍历性。

A new matrix-infinite-product-form solution for upper block-Hessenberg Markov chains and its quasi-algorithmic constructibility

This paper presents a new matrix-infinite-product-form (MIP-form) solution for the stationary distribution in upper block-Hessenberg Markov chains (UBH-MCs). The existing MIP-form solution (Masuyama, Queueing Syst., Vol. 92, 2019, pp. 173--200) requires a certain parameter set that satisfies both a Foster-Lyapunov drift condition and a convergence condition. In contrast, the new MIP-form solution requires no such parameter sets and no other conditions. The new MIP-form solution also has "quasi-algorithmic constructibility", which is a newly introduced feature of being constructed by iterating …

Subexponential upper and lower bounds in Wasserstein distance for Markov processes

In this article, relying on Foster-Lyapunov drift conditions, we establish subexponential upper and lower bounds on the rate of convergence in the $\mathrm{L}^p$-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We further discuss these results in the context of Markov Lévy-type processes. In the lack of irreducibility and/or aperiodicity properties, we obtain exponential ergodicity in the $\mathrm{L}^p$-Wasserstein distance for a class of Itô processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specif…

Specific bounds for a probabilistically interpretable solution of the Poisson equation for general state-space Markov chains with queueing applications

This paper considers the Poisson equation for general state-space Markov chains in continuous time. The main purpose of this paper is to present specific bounds for the solutions of the Poisson equation for general state-space Markov chains. The solutions of the Poisson equation are unique in the sense that they are expressed in terms of a certain probabilistically interpretable solution (called the {\it standard solution}). Thus, we establish some specific bounds for the standard solution under the $f$-modulated drift condition (which is a kind of Foster-Lyapunov-type condition) and some mode…

非Feller马尔可夫半群的不变测度的存在性和唯一性

Existence and uniqueness of invariant measures for non-Feller Markov semigroups

本文研究非Feller马尔可夫半群不变测度的存在性和唯一性，通过测度论方法证明在自然假设下唯一性成立。

Full classification of dynamics for one-dimensional continuous time Markov chains with polynomial transition rates

This paper provides full classification of dynamics for continuous time Markov chains (CTMCs) on the non-negative integers with polynomial transition rate functions. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence vs transience, certain absorption, positive recurrence vs null recurrence, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary di…

Some new results on sample path optimality in ergodic control of diffusions

We present some new results on sample path optimality for the ergodic control problem of a class of non-degenerate diffusions controlled through the drift. The hypothesis most often used in the literature to ensure the existence of an a.s. sample path optimal stationary Markov control requires finite second moments of the first hitting times $τ$ of bounded domains over all admissible controls. We show that this can be considerably weakened: ${\mathrm E}[τ^2]$ may be replaced with ${\mathrm E}[τ\ln^+(τ)]$, thus reducing the required rate of convergence of averages from polynomial to logarithmic…

Mixing times for two classes of stochastically modeled reaction networks

The past few decades have seen robust research on questions regarding the existence, form, and properties of stationary distributions of stochastically modeled reaction networks. When a stochastic model admits a stationary distribution an important practical question is: what is the rate of convergence of the distribution of the process to the stationary distribution? With the exception of \cite{XuHansenWiuf2022} pertaining to models whose state space is restricted to the non-negative integers, there has been a notable lack of results related to this rate of convergence in the reaction network…

Rank-driven Markov processes

We study a class of Markovian systems of $N$ elements taking values in $[0,1]$ that evolve in discrete time $t$ via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak--Sneppen model of evolution, in which the system represents an evolutionary 'fitness landscape' and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-$N$ asymptotics for the general model in which, at each time step, $K$ randomly chosen elements are discarded and replaced by inde…

欠阻尼 Langevin 动力学离散化的一致小化条件与收敛界

Uniform minorization condition and convergence bounds for discretizations of kinetic Langevin dynamics

本文研究欠阻尼 Langevin 动力学多种离散化格式的收敛性，证明关于步长一致的小化与 Foster–Lyapunov 漂移条件，并得到随步长线性缩放的几何收敛界。