Particle systems with quasi-homogeneous initial states and their occupation time fluctuations

Occupation times of subcritical branching immigration systems with Markov motion, clt and deviations principles

In this paper we consider two related stochastic models. The first one is a branching system consisting of particles moving according to a Markov family in R^d and undergoing subcritical branching with a constant rate of V>0. New particles immigrate to the system according to a homogeneous space time Poisson random field. The second model is the superprocess corresponding to the branching particle system. We study rescaled occupation time process and the process of its fluctuations with very mild assumptions on the Markov family. In the general setting a functional central limit theorem as wel…

分支马尔可夫链的拟过程

Quasi-processes for branching Markov chains

本文针对分支马尔可夫链构建位势理论的概率对应形式，提出分支拟过程，证明其与过度测度的同构关系，构造带分支的随机交错结构，以分支过程脊表示为关键技术工具。

Markov processes with spatial delay: path space characterization, occupation time and properties

In this paper, we study one dimensional Markov processes with spatial delay. Since the seminal work of Feller, we know that virtually any one dimensional, strong, homogeneous, continuous Markov process can be uniquely characterized via its infinitesimal generator and the generator's domain of definition. Unlike standard diffusions like Brownian motion, processes with spatial delay spend positive time at a single point of space. Interestingly, the set of times that a delay process spends at its delay point is nowhere dense and forms a positive measure Cantor set. The domain of definition of the…

Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses

The $(d,α,β,γ)$-branching particle system consists of particles moving in $R^d$ according to a symmetric $α$-stable Lévy process $(0<α\leq 2)$, splitting with a critical $(1+β)$-branching law $(0<β\leq 1)$, and starting from an inhomogeneous Poisson random measure with intensity measure $μ_γ(dx)=dx/(1+|x|^γ), γ\geq 0$. By means of time rescaling $T$ and Poisson intensity measure $H_Tμ_γ$, occupation time fluctuation limits for the system as $T\to\infty$ have been obtained in two special cases: Lebesgue measure ($γ=0$, the homogeneous case), and finite measures $(γ>d)$. In some cases $H_T\equiv…

Asymptotic property of the occupation measures in a two-dimensional skip-free Markov modulated random walk

We consider a discrete-time two-dimensional process $\{(X_{1,n},X_{2,n})\}$ on $\mathbb{Z}^2$ with a background process $\{J_n\}$ on a finite set $S_0$, where individual processes $\{X_{1,n}\}$ and $\{X_{2,n}\}$ are both skip free. We assume that the joint process $\{\boldsymbol{Y}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(X_{1,n},X_{2,n})\}$ vary according to the state of the background process $\{J_n\}$. This modulation is assumed to be space homogeneous. We refer to this process as a two-dimensional skip-free Markov …

Asymptotic property of the occupation measures in a multi-dimensional skip-free Markov modulated random walk

We consider a discrete-time $d$-dimensional process $\{\boldsymbol{X}_n\}=\{(X_{1,n},X_{2,n},...,X_{d,n})\}$ on $\mathbb{Z}^d$ with a background process $\{J_n\}$ on a countable set $S_0$, where individual processes $\{X_{i,n}\},i\in\{1,2,...,d\},$ are skip free. We assume that the joint process $\{\boldsymbol{Y}_n\}=\{(\boldsymbol{X}_n,J_n)\}$ is Markovian and that the transition probabilities of the $d$-dimensional process $\{\boldsymbol{X}_n\}$ vary according to the state of the background process $\{J_n\}$. This modulation is assumed to be space homogeneous. We refer to this process as a $…

The exit time finite state projection scheme: bounding exit distributions and occupation measures of continuous-time Markov chains

We introduce the exit time finite state projection (ETFSP) scheme, a truncation-based method that yields approximations to the exit distribution and occupation measure associated with the time of exit from a domain (i.e., the time of first passage to the complement of the domain) of time-homogeneous continuous-time Markov chains. We prove that: (i) the computed approximations bound the measures from below; (ii) the total variation distances between the approximations and the measures decrease monotonically as states are added to the truncation; and (iii) the scheme converges, in the sense that…