Poisson Hypothesis for Information Networks (A study in non-linear Markov processes) I. Domain of Validity

Ergodic control of diffusions with compound Poisson jumps under a general structural hypothesis

We study the ergodic control problem for a class of controlled jump diffusions driven by a compound Poisson process. This extends the results of [SIAM J. Control Optim. 57 (2019), no. 2, 1516-1540] to running costs that are not near-monotone. This generality is needed in applications such as optimal scheduling of large-scale parallel server networks. We provide a full characterization of optimality via the Hamilton-Jacobi-Bellman (HJB) equation, for which we additionally exhibit regularity of solutions under mild hypotheses. In addition, we show that optimal stationary Markov controls are a.s.…

The Bell-Touchard Counting process

The Poisson process is one of the simplest stochastic processes defined in continuous time, having interesting mathematical properties, leading, in many situations, to applications mathematically treatable. One of the limitations of the Poisson process is the rare events hypothesis; which is the hypothesis of unitary jumps within an infinitesimal window of time. Although that restriction may be avoided by the compound Poisson process, in most situations, we don't have a closed expression for the probability distribution of the increments of such processes, leaving us options such as working wi…

Replica-Mean-Field Limits of Fragmentation-Interaction-Aggregation Processes

Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called "Poisson Hypothesis". However, in most applications, this hypothesis is only conjectured. Here, we establish the Poisson Hypothe…

Poisson聚类过程两种子模型中泛型聚类的尾渐进行为与精确大偏差分析

Tail asymptotics and precise large deviations for some Poisson cluster processes

本文研究了再生Poisson聚类过程和Hawkes过程的尾渐进行为，利用正则变化性假设和Karamata定理推导出精确的大偏差结果。

Hunt's Hypothesis (H) for the Sum of Two Independent Levy Processes

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.

New results on Hunt's hypothesis (H) for Lévy processes

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.

连续时间分裂-交互-聚合过程的副本-平均场极限

Replica-mean-field limit of continuous-time fragmentation-interaction-aggregation processes

本文引入连续分裂-交互-聚合过程，研究其副本-平均场版本，并证明在无限副本极限下泊松假设成立，为具有点状交互的网络动力学提供了可处理性。

Further study on Hunt's hypothesis (H) for Levy processes

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…

Mixing properties of non-stationary INGARCH(1,1) processes

We derive mixing properties for a broad class of Poisson count time series satisfying a certain contraction condition. Using specific coupling techniques, we prove absolute regularity at a geometric rate not only for stationary Poisson-GARCH processes but also for models with an explosive trend. We provide easily verifiable sufficient conditions for absolute regularity for a variety of models including classical (log-)linear models. Finally, we illustrate the practical use of our results for hypothesis testing.