跳过程的不变测度与q对的不变测度

本文讨论了一般状态空间上具有不变测度的q对的对偶q对的构造问题,证明了正则q对的不变测度是它的相应跳过程的不变测度的充要条件是该q对的对偶q对是正则的。     

一般状态空间跳过程的强遍历性

研究了一般状态空间跳过程的强遍历性,利用最小非负解理论及马氏性,得到了强遍历性的几个等价条件,把连续时间可数状态马氏链的相关结果推广到一般状态空间跳过程的情形.     

跳过程μ正则性和不变测度存在性

本文给出了一般状态跳过程μ正则的充分条件,作为其推论得到跳跃链常返的跳过程是μ正则的,证明了跳跃链常返的跳过程,其q对的不变测度是跳过程的不变测度.还证明了跳跃链常返的跳过程存在唯一的不变测度.     

一般状态空间跳过程的Harris常返

研究了一般状态空间跳过程的Harris常返,利用马氏性,得到了跳过程Harris常返的几个等价条件.     

跳过程的强遍历性

研究了一般状态空间跳过程的强遍历性,利用骨架链的方法,给出了强遍历性的几个等价条件,得到的结论类似于离散时间一般状态空间马氏链.     

跳过程存在成功耦合的充要条件

本文研究了一般状态跳过程的ｈ骨架的不变σ代数，尾σ－代数之间的关系，并由此证明具有正则ｑ对的非常返跳过程存在成功耦合的充要条件是跳过程的所有有界调和函数都是常数。     

一般状态空间跳过程的遍历性

该文研究了一般状态空间跳过程的遍历性,得到了与连续时间可数状态空间马氏链类似的结果.     

肖争艳等： 绕积马氏链的状态分类

该文给出了绕积马氏链的特征数和状态的定义, 利用一般马氏链的理论讨论了随机环 境中的马氏链的各种状态的特征以及各类状态之间的联系, 还给出了在联合空间不可分解且 正则本质的条件下, 状态正则本质的充要条件. 最后举例说明了经典马氏链和随机环境中马氏链的状态的区别.     

一般状态空间跳过程不变测度的存在性和唯一性

本文利用一般状态空间马氏链关于不变测度的有关结果和跳过程的性质，研究了一般状态空间跳过程不变测度的存在性和唯一性。     

一般状态空间跳过程的不可约性

本文研究了一般状态空间跳过程的ψ-不可约性.利用与马氏链类似的方法,得到了最大不可约测度及其性质,推广了离散时间一般状态空间马氏链的结果,作为应用,给出了C∈E+的充分条件.最后,证明了跳过程、骨架链、跳跃链和预解链的ψ-不可约是等价的.     

Extremal behavior of regularly varying stochastic processes

We study a formulation of regular variation for multivariate stochastic processes on the unit interval with sample paths that are almost surely right-continuous with left limits and we provide necessary and sufficient conditions for such stochastic processes to be regularly varying. A version of the Continuous Mapping Theorem is proved that enables the derivation of the tail behavior of rather general mappings of the regularly varying stochastic process. For a wide class of Markov processes with increments satisfying a condition of weak dependence in the tails we obtain simplified sufficient conditions for regular variation. For such processes we show that the possible regular variation limit measures concentrate on step functions with one step, from which we conclude that the extremal behavior of such processes is due to one big jump or an extreme starting point. By combining this result with the Continuous Mapping Theorem, we are able to give explicit results on the tail behavior of various vectors of functionals acting on such processes. Finally, using the Continuous Mapping Theorem we derive the tail behavior of filtered regularly varying Lévy processes.     

Branching random walks,stable point processes and regular variation

Using the theory of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is used to obtain the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. Because of heavy tails of the step size distribution, we can invoke a one large jump principle at the level of point processes to give an explicit representation of the limiting point process. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.     

The limit distribution of the maximum increment of a random walk with dependent regularly varying jump sizes

We investigate the maximum increment of a random walk with heavy-tailed jump size distribution. Here heavy-tailedness is understood as regular variation of the finite-dimensional distributions. The jump sizes constitute a strictly stationary sequence. Using a continuous mapping argument acting on the point processes of the normalized jump sizes, we prove that the maximum increment of the random walk converges in distribution to a Fréchet distributed random variable.     

A thinning result for pure jump processes

A standard thinning procedure for point processes is extended to processes of pure jump type in which each jump is retained with probability p or deleted with probability 1 ? p, independently of everything else.Two theorems are proved, the first gives a sufficient condition for the existence of thinned pure jump processes, the second concerns the convergence of such processes to pure jump processes whose increments are generated by a Cox process. Some generalizations are discussed.     

Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory

We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto–Piepenbrink type theorem, which is based on a ground state transform, and a Shnol' type theorem. Our setting includes Laplacian on manifolds, on graphs and α-stable processes.     

Asymptotic equivalence for nonparametric regression with non-regular errors

Asymptotic equivalence in Le Cam’s sense for nonparametric regression experiments is extended to the case of non-regular error densities, which have jump discontinuities at their endpoints. We prove asymptotic equivalence of such regression models and the observation of two independent Poisson point processes which contain the target curve as the support boundary of its intensity function. The intensity of the point processes is of order of the sample size n and involves the jump sizes as well as the design density. The statistical model significantly differs from regression problems with Gaussian or regular errors, which are known to be asymptotically equivalent to Gaussian white noise models.     

Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces

We give equivalent characterizations for off-diagonal upper bounds of the heat kernel of a regular Dirichlet form on the metric measure space, in two settings: for the upper bounds with the polynomial tail (typical for jump processes) and for the upper bounds with the exponential tail (for diffusions). Our proofs are purely analytic and do not use the associated Hunt process.