Random Walks on Trees with Finitely Many Cone Types

This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and homesick random walks. We give a simple criterion for transience or recurrence of the random walk and prove that the spectral radius is equal to 1 if and only if the random walk is recurrent. Furthermore, we study the asymptotic behaviour of return probabilitites and prove a local limit theorem. In the transient case, we also prove a law of large numbers and compute the rate of escape of the random walk to infinity, as well as prove a central limit theorem. Finally, we describe the structure of the boundary process and explain its connection with the random walk.     

Bivariate step processes and random evolutions

A random evolution process constructed from regular step processes with a common state space and indexed on an evolution rule space is shown to be a regular step process on the product space. Conversely, it is shown that under mild conditions, any regular step process on a product space is equivalent to a random evolution process. Conditions are given on the cardinality of the spaces and on the parameters of the process that are sufficient for the process to have various recurrence and ergodicity properties. Applications to birth-death processes are given.     

Tubular recurrence

We introduce the directed-edge-reinforced random walk and prove that the process is equivalent to a random walk in random environment. Using Oseledec"s multiplicative ergodic theorem, we obtain recurrence and transience criteria for random walks in random environment on graphs with a certain linear structure and apply them to directed-edge-reinforced random walks. This revised version was published online in August 2006 with corrections to the Cover Date.     

Frogs in Random Environment

We study the so-called frog model: Initially there are some sleeping particles and one active particle. A sleeping particle is activated when an active particle hits it, after that the activated particle starts to walk independently of everything and can activate other sleeping particles as well. The initial configuration of sleeping particles is random with density p(x). We identify the critical rate of decay of p(x) separating transience from recurrence, and study some other properties of the model.     

Degrees of Transience and Recurrence and Hierarchical Random Walks

The notion of degree and related notions concerning recurrence and transience for a class of Lévy processes on metric Abelian groups are studied. The case of random walks on a hierarchical group is examined with emphasis on the role of the ultrametric structure of the group and on analogies and differences with Euclidean random walks. Applications to separation of time scales and occupation times of multilevel branching systems are discussed. Mathematics Subject Classifications (2000) 60G50, 60B15, 60F05, 60J80.D.A. Dawson: Research supported by NSERC (Canada) and a Max Planck Award for International Cooperation.L.G. Gorostiza: Research supported by CONACYT grant 37130-E (Mexico).A. Wakolbinger: Research supported by DFG (SPP 1033) (Germany).     

Random walks in two-dimensional complexes

A 2-dimensional complex is a union of a finite number of quarter planes ₊ ² having some boundaries in common. The most interesting example is the union of all 2-dimensional faces of ₊ ^N . We consider maximally homogeneous random walks on such complexes and obtain necessary and sufficient conditions for ergodicity, null recurrence and transience up to some non-zero assumptions which are of measure 1 in the parameter space.The problem we address in this paper is of theoretical range. However, the results can be applied to performance evaluation of some telecommunication systems (e.g. local area networks) viewed as interacting queues. To enforce this assertion, a detailed example of coupled queues in differentregimes is presented.     

On transience of Lévy-type processes

In this paper, we study weak and strong transience of a class of Feller processes associated with pseudo-differential operators, the so-called Lévy-type processes. As a main result, we derive Chung-Fuchs type conditions (in terms of the symbol of the corresponding pseudo-differential operator) for these properties, which are sharp for Lévy processes. Also, as a consequence, we discuss the weak and strong transience with respect to the dimension of the state space and Pruitt indices, thus generalizing some well-known results related to elliptic diffusion and stable Lévy processes. Finally, in the case when the symbol is radial (in the co-variable) we provide conditions for the weak and strong transience in terms of the Lévy measures.     

Excessiveness of harmonic functions for certain diffusions

In an earlier paper⁽⁴⁾ the author has shown that a diffusion process whose potential kernel satisfies certain analytic conditions has all of its excessive harmonic functions, which are not identically infinite, continuous. This paper shows that under these conditions (concerning its potential kernel), the excessiveness of its nonnegative harmonic functions isautomatic.     

Recurrence and transience properties of some neural networks: an approach via fluid limit models

The subject of the paper is the stability analysis of some neural networks consisting of a finite number of interacting neurons. Following the approach of Dai [5] we use the fluid limit model of the network to derive a sufficient condition for positive Harris-recurrence of the associated Markov process. This improves the main result in Karpelevich et al. [11] and, at the same time, sheds some new light on it. We further derive two different conditions that are sufficient for transience of the state process and illustrate our results by classifying some examples according to positive recurrence or transience. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some Limit Results for Transient Subsequence of Brownian Motion on Line

Ｌｅｔ {Ｗ(ｔ) ,0≤ｔ<∞ }ｂｅａｓｔａｎｄａｒｄ ,ｏｎｅ ｄｉｍｅｎｓｉｏｎａｌＢｒｏｗｎｉａｎｍｏｔｉｏｎｏｎ (Ω ,Ｆ ,Ｐ) .Ｉｔｉｓｗｅｌｌｋｎｏｗｎｔｈａｔ -∞ =ｌｉｍｉｎｆｔ→∞ Ｗ(ｔ) <ｌｉｍｓｕｐｔ→∞ Ｗ(ｔ) =∞ａｎｄａｃｃｏｒｄｉｎｇｔｏＫａｈａｎｃｅ ([1 ] ,Ｔｈｅｏｒｅｍ1 ,Ｃｈａｐｔｅｒ 1 2 ) ,ｉｆａｓｅｑｕｅｎｃｅ {ｔｎ,ｎ≥ 1 )ｓａｔｉｓｆｉｅｓ∑∞ｎ=11ｔｎ<∞ ,ｔｈｅｎｌｉｍｎ→∞ Ｗ(ｔｎ) =∞ａ .ｓ.Ｗｅｃａｌｌｔｈｅｓｅｑｕｅｎｃｅ {Ｗ (ｔｎ) ,ｎ≥ 1 }ａｔｒａｎｓｉｅｎｔ…