Ze-Chun Hu, Wei Sun
The goal of this paper is threefold. First, we survey the existing results on Hunt's hypothesis (H) for Markov processes and Getoor's conjecture for Lévy processes. Second, we investigate (H) for multidimensional Lévy processes from the viewpoints of projections and energy, respectively. Third, we present a few open questions for further study.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we will consider the problem that how far from Hunt's hypothesis (H) to symmetrization for a general 1-dimensional diffusion. A characterization of (H) involving the classification of points for this diffusion will be first obtained. Then the main result shows that such a process is symmetrizable, if and only if (H) holds and a certain family of asymmetric shunt points is empty. Furthermore, we will also derive the representation of associated Dirichlet forms of general 1-dimensional diffusions under symmetry.
Which Levy processes satisfy Hunt's hypothesis (H) is a long-standing open problem in probabilistic potential theory. The study of this problem for one-dimensional Levy processes suggests us to consider (H) from the point of view of the sum of Levy processes. In this paper, we present theorems and examples on the validity of (H) for the sum of two independent Levy processes. We also give a novel condition on the Levy measure which implies (H) for a large class of one-dimensional Levy processes.
Hunt's hypothesis (H) and the related Getoor's conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt's hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes $(X_t)$ and $(Y_t)$, if $(X_t)$ satisfies (H) and $(Y_t)$ is locally absolutely continuous with respect to $(X_t)$, then $(Y_t)$ satisfies (H). Our second theorem shows that a standard process $(X_t)$ satisfies (H) if and only if $…
In this paper, we present new results on Hunt's hypothesis (H) for Lévy processes. We start with a comparison result on Lévy processes which implies that big jumps have no effect on the validity of (H). Based on this result and the Kanda-Forst-Rao theorem, we give examples of subordinators satisfying (H). Afterwards we give a new necessary and sufficient condition for (H) and obtain an extended Kanda-Forst-Rao theorem. By virtue of this theorem, we give a new class of Lévy processes satisfying (H). Finally, we construct a type of subordinators that does not satisfy Rao's condition.
In this paper, Hunt's hypothesis (H) and Getoor's conjecture for Lévy processes are revisited. Let $X$ be a Lévy process on $\mathbf{R}^n$ with Lévy-Khintchine exponent $(a,A,μ)$. {First, we show that if $A$ is non-degenerate then $X$ satisfies (H). Second, under the assumption that $μ({\mathbf{R}^n\backslash \sqrt{A}\mathbf{R}^n})<\infty$, we show that $X$ satisfies (H) if and only if the equation $$ \sqrt{A}y=-a-\int_{\{x\in {\mathbf{R}^n\backslash \sqrt{A}\mathbf{R}^n}:\,|x|<1\}}xμ(dx),\ y\in \mathbf{R}^n, $$ has at least one solution. Finally, we show that if $X$ is a subordinator and sati…
Getoor's conjecture that essentially all Levy processes satisfy (H) is a long-standing open problem in potential theory. In the beginning of the paper, we summarize the main results obtained so far for the problem. Then, we present two new necessary and sufficient conditions for the validity of (H). Furthermore, we give applications of these new criteria. First, we give explicit constructions of Levy processes satisfying (H) in a context where previously known results could not be applied. Second, we show that a large class of pure jump subordinators can be decomposed into the summation of two…
Variational formulas for the Laplace transform of the exit time from an open set of a Hunt process generated by a regular lower bounded semi-Dirichlet form are established. While for symmetric Markov processes, variational formulas are derived for the exponential moments of the exit time. As applications, we provide some comparison theorems and quantitative relations of the exponential moments and Poincaré inequalities.
Construction of Hunt processes by the Lyapunov method and applications to generalized Mehler semigroups
本文通过李雅普诺夫方法结合局部Feller型正则性条件，从一般状态空间的马尔可夫半群构造出Hunt过程，解决了广义Mehler半群对应càdlàg过程的公开问题，并将结果应用于带非对角Lévy噪声的随机热方程。
Let $(S,ρ)$ be an ultrametric space with certain conditions and $S^k$ be the quotient space of $S$ with respect to the partition by balls with a fixed radius $ϕ(k)$. We prove that, for a Hunt process $X$ on $S$ associated with a Dirichlet form $(\mathcal E, \mathcal F)$, a Hunt process $X^k$ on $S^k$ associated with the averaged Dirichlet form $(\mathcal E^k, \mathcal F^k)$ is Mosco convergent to $X$, and under certain additional conditions, $X^k$ converges weakly to $X$. Moreover, we give a sufficient condition for the Markov property of $X$ to be preserved under the canonical projection $π^k…
We establish weak and strong law of large numbers for a class of branching symmetric Hunt processes with the branching rate being a smooth measure with respect to the underlying Hunt process, and the branching mechanism being general and state-dependent. Our work is motivated by recent work on strong law of large numbers for branching symmetric Markov processes by Chen-Shiozawa [J. Funct. Anal., 250, 374--399, 2007] and for branching diffusions by Engländer-Harris-Kyprianou [Ann. Inst. Henri Poincaré Probab. Stat., 46, 279--298, 2010]. Our results can be applied to some interesting examples th…
We treat the class of universal Markov processes on the d-dimensional Euklidean space which do not depend on random. For these, as well as for several subclasses, we prove criteria whether a function f, defined on the positive half-line, can be a path of a process in the respective class. This is useful in particular in the construction of (counter-)examples. Furthermore we characterize the processes of this kind, which are homogeneous in space and time. The semimartingale property is characterized in terms of the jumps of a one-dimensional deterministic Markov process. We emphasize the differ…
The Computation of the Mean First Passage Times for Markov Chains
本文综述了计算马尔可夫链平均首次通过时间的各种方法，提出了一种新的精确计算技术，并与其他方法进行了比较。
We establish the equivalence of the analytic and probabilistic notions of subharmonicity in the framework of general symmetric Hunt processes on locally compact separable metric spaces, extending an earlier work of the first named author on the equivalence of the analytic and probabilistic notions of harmonicity. As a corollary, we prove a strong maximum principle for locally bounded finely continuous subharmonic functions in the space of functions locally in the domain of the Dirichlet form under some natural conditions.
Large deviations related to Dynkin--Lamperti arcsine laws for last visit times of Markov processes
本文建立了从属子的Dynkin-Lamperti反正弦律相关大偏差估计，并将其应用于Hunt过程最后访问时间的大偏差研究，覆盖了多类过程并拓展了相关概率理论结果。
This paper investigates change point detection in state space models, in which the pre-change distribution $f^{θ_0}$ is given, while the poster distribution $f^θ$ after change is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from $f^{θ_0}$ to $f^θ$, under a restriction on the false alarms. We investigate theoretical properties of a weighted Shiryayev-Roberts-Pollak (SRP) change point detection rule in state space models. By making use of a Markov chain representation for the likelihood function, exponential embedding of the induced Markovian trans…
Generalized Kac's moment formula for positive continuous additive functionals of Markov processes
本文推导了马尔可夫过程正连续加法函数的广义Kac动量公式，适用于具有绝对连续转移函数的标准过程及对称Hunt过程。
Applying quantitative perturbation theory for linear operators, we prove non-asymptotic limit theorems for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions X . The main results are concentration inequalities and Berry-Esseen bounds, obtained assuming neither reversibility nor `warm start' hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform X-ergodicity hypothesis, and when X consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the…
In this paper we prove the Poisson Hypothesis for the limiting behavior of the large queueing systems in some simple ("mean-field") cases. We show in particular that the corresponding dynamical systems, defined by the non-linear Markov processes, have a line of fixed points which are global attractors. To do this we derive the corresponding non-linear integral equation and we explore its self-averaging properties. Our derivation relies on a solution of a combinatorial problem of rode placements.
In this paper we study the Poisson Hypothesis, which is a device to analyze approximately the behavior of large queueing networks. We prove it in some simple limiting cases. We show in particular that the corresponding dynamical system, defined by the non-linear Markov process, has a line of fixed points which are global attractors. To do this we derive the corresponding non-linear equation and we explore its self-averaging properties. We also argue that in cases of havy-tail service times the PH can be violated.
In this paper, we investigate the probabilistic characteristics of a unit driven by Dichotomous Markov Noise (DMN), as an external random life increasing and decreasing shocks. Using DMN, we will define two new aging classes of the overall increasing/decreasing (OIL/ODL) nature in the long time behavior, which are separated by an exponential steady state regime. In addition, a moment inequality is derived for the system whose life distribution is in an overall life decreasing (ODL) class. We use this inequality to devise a nonparametric testing procedure for exponentiality against an alternati…
Finite-length Analysis on Tail probability for Markov Chain and Application to Simple Hypothesis Testing
本文利用信息几何推导有限长度Markov链样本均值尾概率及累积量生成函数的界，并据此建立简单假设检验中第二类错误最小概率的Hoeffding型界，还给出中心极限定理的新证明。
Consider an intersection measure $\ell_t ^{\mathrm{IS}}$ of $p$ independent (possibly different) $m$-symmetric Hunt processes up to time $t$ in a metric measure space $E$ with a Radon measure $m$. We derive a Donsker-Varadhan type large deviation principle for the normalized intersection measure $t^{-p}\ell_t ^{\mathrm{IS}}$ on the set of finite measures on $E$ as $t \rightarrow \infty$, under the condition that $t$ is smaller than life times of all processes. This extends earlier work by W. König and C. Mukherjee (2013), in which the large deviation principle was established for the intersect…
In this paper we investigate the behavior of the bridges of a Markov counting process in several directions. We first characterize convexity(concavity) in time of the mean value in terms of lower (upper) bounds on the so called \textit{reciprocal characteristics}. This result gives a natural criterion to determine whether bridges are "lazy" or "hurried". Under the hypothesis of global bounds on the reciprocal characteristics we prove sharp estimates for the marginal distributions and a comparison theorem for the jump times. When the height of the bridge tends to infinity we show the convergenc…
Stable processes with reflection
本文构造了从有界开 Lipschitz 集合补集反射的各向同性 α-稳定 Lévy 过程，提出基于非局部 Schrödinger 扰动和超中值函数的 Markov 过程拼接新方法，并引入记录反射次数的阶梯过程以刻画返回分布与平稳分布。
Let X and Y be time-homogeneous Markov processes with common state space E, and assume that the transition kernels of X and Y admit densities with respect to suitable reference measures. We show that if there is a time t>0 such that, for each x\in E, the conditional distribution of (X_s)_{0 < s < t}, given X_0 = x = X_t, coincides with the conditional distribution of (Y_s)_{0 < s < t}, given Y_0 = x = Y_t, then the infinitesimal generators of X and Y are related by [L^Y]f = ψ^{-1}[L^X](ψf)-λf, where ψis an eigenfunction of L^X with eigenvalue λ. Under an additional continuity hypothesis, the s…
We consider a large family of discrete and continuous time controlled Markov processes and study an ergodic risk-sensitive minimization problem. Under a blanket stability assumption, we provide a complete analysis to this problem. In particular, we establish uniqueness of the value function and verification result for optimal stationary Markov controls, in addition to the existence results. We also revisit this problem under a near-monotonicity condition but without any stability hypothesis. Our results also include policy improvement algorithms both in discrete and continuous time frameworks.
Information divergences of Markov chains and their applications
本文定义有限马尔可夫链转移矩阵上的信息散度，推导其 Pinsker 不等式等性质，并用于二元假设检验、Cesàro 混合时间与遍历性系数分析。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。