Joanna B. Mitro

The type problem for random walks on trees

We consider the question of whether the simple random walk (SRW) on an infinite tree is transient or recurrent. For randomℕ-trees (all vertices of distancen from the root of the tree have degreedn, where {dn} are independent random variables), we prove that the SRW is a.s. transient if lim infn→∞nE(log(dn-1))>1 and a.s. recurrent if lim supn→∞n E(log(dn-1))<1. For random trees in which the degrees of the vertices are independently 2 or 3, with distribution depending on the distance from the root, a partial classification of type is obtained.

Applications of Revuz and Palm Type Measures for Additive Functionals in Weak Duality

Several characterizations of additive functionals of a Markov process have been described in recent years. Under strong (Hunt) duality hypotheses this was accomplished in a series of papers by Revuz [14], [15], Getoor [9], and Sharpe [17]; for “symmetric” processes this was done by Fukushima [7] and Dynkin [4], [5]; earlier, the situation for Markov stochastic systems was investigated by Dynkin [3], [6]. Here, we obtain results along the same lines for processes in weak duality. The main tool is the “auxiliary process” [13] associated to a pair of Markov processes in weak duality. (Some facts about this process are recalled below.) Our approach is guided in part by similarities with the theory of flows ([8], [16]) and exploits the interplay between optionality and cooptionality in this context.

A discontinuous time change for natural additive functionals which preserves duality

SummaryContinuous time changes of Markov processes preserve duality, but a discontinuous time change recently proposed by Weidenfeld does not. We modify his procedure to obtain a time change which preserves duality when the time changing functional is natural.

Time reversal depending on local time

The process (X, l), where X is a Markov process and l its local time at a regular point b, is reversed from the time l first exceeds the level t, and the resulting process is identified under duality hypotheses. The approach exploits recent results in the theory of excursions of dual processes.

General theory of markov processes

General theory of markov processes, by Michael Sharpe, University of California at San Diego. Academic Press, New York (1988), 419 pp.

Discontinuous Time Changes and Duality for Markov Processes

49.50. ISBN 0-12-639060-6.

Dual Markov processes: Construction of a useful auxiliary process

The purpose of this paper is to continue work on discontinuous time changes of dual Markov processes begun in [6], where time changes based on discontinuous natural additive functionals were discussed. Both continuous and discontinuous natural time changes share the features (i) dual time changes (i.e., based on dual additive functionals) preserve duality, (ii) the new duality measure of the time changed processes is constructed from the Revuz measure of the time changing additive functionals. In this paper, we consider the general case of time changes based on additive functionals which have a quasi-left-continuous purely discontinuous component, and present for them a time changing procedure which possesses features (i) and (ii) above.