Degrees of Transience and Recurrence and Hierarchical Random Walks

Abstract

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. (2000): 60J15, 60J30, 60B15, 60F05, 60J80. Mathematics Subject Classifications

Hierarchical Equilibria of Branching Populations

“…and a necessary condition is that the degree of transience be equal to or greater than 1 [DGW2]. Moreover, as mentioned above simple symmetric d-dimensional random walk has degree of transience d/2 − 1, and it is strongly transient iff d > 4.…”

“…The migration specified above is a (continuous time) random walk on Ω N called hierarchical random walk. Hierarchical random walks, in particular their transience and recurrence properties, are studied in [DGW2], [DGW3].…”

“…The following proposition is proved in [DGW2]. ) denotes a sequence of nested balls of radii ℓ in Ω N ).…”

“…where p t is the transition probability of Z [DGW2,SaW]. For simple symmetric random walk on the d-dimensional lattice Z d , it is wellknown that dimension 2 is the borderline for transience.…”

Percolation in a Hierarchical Lattice

“…The random walks in Ω N can go far only by means of long-range jumps, which is clearly not the case in Z d ; c.f. [19,23]. Now consider a long-range percolation on Ω N .…”