Paolo M. Soardi

Potential Theory on Infinite Networks

Kirchhoffs laws.- Finite networks.- Currents and potentials withfinite energy.- Uniqueness and related topics.- Some examples and computations.- Roydens compactification.- Rough isometries.

Existence of fixed points of nonexpansive mappings in certain Banach lattices

A fixed point theorem for nonexpansive mappings in dual Banach spaces is proved. Applications in certain Banach lattices are given. 1. Suppose K is a subset of a Banach space X and T: K -* K is a nonexpansive mapping, i.e. II T(x) T(y)jj S lix -yIj, x, y E K. A wellknown theorem due to Kirk [1] states that, if K is convex weakly compact (weak* compact when X is a dual space) and has normal structure, then T has a fixed point in K. In particular, if X = LP (1 0. Here and in the sequel V and A denote the least upper bound and the greatest lower bound respectively. X is said to be order complete if each set A C X with an upper bound has a least upper bound. A complex AM-space is defined as the complexification of an AM-space. Suppose X is an order complete AM-space with unit (i.e. an element e such that the unit ball at zero is the order interval [e, e]); then X is isometrically lattice isomorphic to the space CR (S) of all continuous real-valued functions defined on a compact Stonian space S. For these and other facts about Banach lattices we refer to Schaefers book [4]. Received by the editors March 19, 1978. AMS (MOS) subject classifications (1970). Primary 47H10; Secondary 46B99, 46E05.

p-harmonic functions on graphs and manifolds.

We show that the LiouvilleDp-property is invariant under rough isometries between a Riemannian manifold of bounded geometry and a graph of bounded degree.

Rough isometries and Dirichlet finite harmonic functions on graphs

Suppose that G 1 and G 2 are roughly isometric connected graphs of bounded degree. If G 1 has no nonconstant Dirichlet finite harmonic functions, then neither has G 2

Random walks on free products, quotients and amalgams

Suppose that G is a discrete group and p is a probability measure on G . Consider the associated random walk {X n } on G . That is, let X n = Y 1 Y 2 … Y n , where the Y j ’s are independent and identically distributed G -valued variables with density p . An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p . For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x ) permits a description of the asymptotic behaviour of these probabilities.

Martin and end compactifications for non-locally finite graphs

We consider a connected graph, having countably infinite vertex set X, which is permitted to have vertices of infinite degree. For a transient irreducible transition matrix P corresponding to a nearest neighbor random walk on X, we study the associated harmonic functions on X and, in particular, the Martin compactification. We also study the end compactification of the graph. When the graph is a tree, we show that these compactifications coincide; they are a disjoint union of X, the set of ends, and the set of improper vertices-new points associated with vertices of infinite degree. Other results proved include a solution of the Dirichlet problem in the context of the end compactification of a general graph. Applications are given to, e.g., the Cayley graph of a free group on infinitely many- generators

Uniqueness of currents in infinite resistive networks

Abstract If an infinite resistive network, whose edges have resistance 1 ohm, satisfies a certain graph theoretical condition, then the homogeneous Kirchhoff equations have no nonzero solutions vanishing at infinity. Every vertex transitive graph with polynomial growth satisfies such a condition. Furthermore uniqueness holds in Cartesian products of infinite regular graphs. Graphs with more than one end and satisfying an isoperimetric inequality provide a counterexample to uniqueness. These results extend partially also to networks with nonconstant resistances.

Convergence to ends for random walks on the automorphism group of a tree

Let μ be a probability on a free group Γ of rank r ≥2. Assume that Supp(μ) is not contained in a cyclic subgroup of Γ. We show that if (X n )n≥0 is the right random walk on Γ determined by μ, then with probability 1, X n converges (in the natural sense) to an infinite reduced word. The space Ω of infinite reduced words carries a unique probability ν such that (Ω, ν) is a frontier of (Γ, μ) in the sense of Furstenberg. This result extends to the right random walk (X n ) determined by a probability μ on the group G of automorphisms of an arbitrary infinite locally finite tree T

The Plancherel measure for polygonal graphs

SummaryWe compute explicitely the Plancherel measure for groups acting isometrically and simply transitively on polygonal graphs.

Harmonic analysis on the free product of two cyclic groups

Abstract Let G = Z r ∗ Z x = 〈 a, b | a r = b s = e 〉 , where r > s ⩾ 2. A natural length is defined for the elements of G and χ 1 denotes the characteristic function of the set of elements of length 1. We study the convolution C ∗ -algebra A ( χ 1 ) generated by χ 1 . We obtain explicitly the associated Plancherel measure, orthogonal polynomials, and spherical functions, and show that A ( χ 1 ) is not a maximal abelian subalgebra of the reduced group C ∗ -algebra of G . Closely related to work by Figa-Talamanca, Picardello, Pytlik, and others on free groups, this study is complicated by the fact that A ( χ 1 ) does not consist of radial functions.