Sample path behaviour in connection with generalized arcsine laws

Summary LetG=(G(t),t0) be the process of last passage times at some fixed point of a Markov process. The Dynkin-Lamperti theorem provides a necessary and sufficient condition forG(t)/t to converge in law ast to some non-degenerate limit (which is then a generalized arcsine law). Under this condition, we give a simple integral test that characterizes the lower-functions ofG. We obtain a similar result forA ⁺=(A ⁺ (t),t0), the time spent in [0, ) by a real-valued diffusion process, in connection with Watanabe's recent extension of Lévy's second arcsine law.     

关于O-U型马氏过程维数和碰撞的一点结果

本文给出了维O－U型马氏过程像集的Housdorff维数上下界的一个估计，并研究了两个独立的一维O－U型马氏过程的碰撞问题。     

The stochastic fluid–fluid model: A stochastic fluid model driven by an uncountable-state process,which is a stochastic fluid model itself

We introduce the Stochastic Fluid–Fluid Model, which offers powerful modeling ability for a wide range of real-life systems of significance. We first derive the infinitesimal generator, with respect to time, of the driving stochastic fluid model. We then use this to derive the infinitesimal generator of a particular Laplace–Stieltjes transform of the model, which is the foundation of our analysis. We develop expressions for the Laplace–Stieltjes transforms of various performance measures for the transient and limiting analysis of the model. This work is the first direct analysis of a stochastic fluid model that is Markovian on a continuous state space.     

Countably many simultaneous random changes of time scales

Simultaneous changes of time scales of the components of a vector Markov process are defined and developed. Measurability properties, Dynkin's lemma, and the strong Markov property are established for the transformed process.     

Ornstein－Uhlenbeck型马氏过程局部时与占位时的存在性

本文考虑Ｏｒｎｓｔｅｉｎ－Ｕｈｌｅｎｂｅｃｋ型马氏过程的局部时,证明了在一定情形下局部时的存在性,并给出了不存在的反例,同时讨论了这类过程的占位时,指出了在某些限制性条件下,占位时密度的平方可积性．     

ON THE FRACTAL NATURE OF INCREMENTS OF THE INFINITE SERIES OF OU PROCESSES RELATED TO THE CHUNG LIL

The author proves that the set of points where the Chung type LIL fails for the path of the infinite series of independent Ornstein-Uhlenbeck processes is a random fraetal, and evaluates its Hausdorff dimension.     

非退化扩散过程的水平集和极集

设X(t)(t∈R )是一个d维非退化扩散过程．本文得到了比原有结果更一般的非退化扩散过程极性的充分条件,证明了对任意u∈Rd,紧集E(0, ∞),有若d=1,则对任意紧集F(?)R, 若d≥2,则对任意紧集E ∈(0, ∞), 其中B(Rd)为Rd上的Borel σ-代数,dim和Dim分别表示Hausdorff维数和Packing 维数．     

Level-crossing approach to a time-limited service system with two types of vacations

We consider an M^X/G/1 queue with nonpreemptive time-limited service and timer and exhaustive vacations. We analyze the waiting time distribution in this multiple vacation model by applying the level-crossing method to a workload process with two types of vacations.     

Birth delay of a Markov process and the stopping distributions for regular processes

Summary We consider Markov processes with a fixed transition functionp(r, x; t, B) and with random birth times. We show that a process can be obtained from (X _t ,P) by birth delay if and only if for allt andB. As an application, we give a new version and a new proof of the results of Rost [R] and Fitzsimmons [F2] on stopping distributions of Markov processes. The key Lemma 1.1 replaces the filling scheme used by the previous authors.Birth delay was considered from a different prospective in [F1].Partially supported by the National Science Foundation Grant DMS-8802667     

MOMENT OF LEvY MEASURE OF OPERATOR-STABLE LAW

In 1979, Jurek gave a characterization of the moment of a full operator-stable μ by eigenvalues of exponent matrix of μ. Here, a characterization of the moment of Lévy measure (restricted on a neighbor of 0) of a full operator-stable μ by eigenvalues of exponent matrix of μ is given.     

On the occupation times of cones by Brownian motion

Summary We study some features concerning the occupation timeA _t of a d-dimensional coneC by Brownian motion. In particular, in the case whereC is convex, we investigate the asymptotic behaviour ofP(A₁u0, when the Brownian motion starts at the vertex ofC. We also give the precise integral test, which decides whether a.s., lim inf _t A _t/(tf(t))=0 or for a decreasing functionf.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

Lévy processes with no positive jumps at an increase time

Summary We study the behaviour of a Lévy process with no positive jumps near its increase times. Specifically, we construct a local time on the set of increase times. Then, we describe the path decomposition at an increase time chosen at random according to the local time, and we evaluate the rate of escape before and after this instant.     

Constructions of coupling processes for Lévy processes

We construct optimal Markov couplings of Lévy processes, whose Lévy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection.     

Multifractal spectra and precise rates of decay in homogeneous fragmentations

We consider a mass-conservative fragmentation of the unit interval. Motivated by a result of Berestycki [J. Berestycki, Multifractal spectra of fragmentation processes, J. Statist. Phys. 113 (3–4) (2003) 411–430], the main purpose of this work is to specify the Hausdorff dimension of the set of locations having exactly an exponential decay. The study relies on an additive martingale which arises naturally in this setting, and a class of Lévy processes constrained to stay in a finite interval.