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Factorization of the Green’s operator and weak-type estimates for a random walk on a tree. (English) Zbl 0729.60008

Let X be a tree. Assume that the order of the vertices is bounded above and at least three. Consider a nearest neighbor random walk on X. Call its transition probability operator P and assume that it is strongly transient, i.e. starting from a vertex u the probability of non-return to any neighbor v of u is larger than some positive \(\delta\) independent of u. To give an orientation to the edges of the tree and to study the sample paths of the random walk it is convenient to work with two so- called realizations: the disk realization and the half-plane realization of the tree.
Introduce the Laplacian \(\Delta =P-I\) and Green’s operator \(G=\sum^{\infty}_{k=0}P^ k\). The main theme of the paper is the study of these operators on the spaces \(l^ p\) by means of various factorizations, where the factors usually admit a probabilistic interpretation (which may be slightly different in the two realizations). It is shown, e.g., that G is bounded on \(l^ p\), p strictly greater than some \(p_ 0\). \(p_ 0\) may very well be larger than 1. If the random walk is strongly reversible, a notion introduced in P. Gerl [J. Theor. Prob. 1, No.2, 171-187 (1988; Zbl 0639.60072)], then \(p_ 0=1.\)
Fundamental references in the field include A. Korányi, M. A. Picardello and M. H. Taibleson [Harmonic analysis, symmetric spaces and probability theory, Cortona/Italy 1984, Symp. Math. 29, 205- 254 (1987; Zbl 0637.31004)] and P. Cartier [Symp. Math. 9, Calcolo Probab., Teor. Turbolenza 1971, 203-270 (1972; Zbl 0283.31005)] as well as the cited paper by Gerl.
Reviewer: G.Högnäs (Åbo)

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
05C05 Trees
43A99 Abstract harmonic analysis
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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