Maura Salvatori

On the norms of group-invariant transition operators on graphs

In this paper we consider reversible random walks on an infinite grapin, invariant under the action of a closed subgroup of automorphisms which acts with a finite number of orbits on the vertex-set. Thel2-norm (spectral radius) of the simple random walk is equal to one if and only if the group is both amenable and unimodular, and this also holds for arbitrary random walks with bounded invariant measure. In general, the norm is bounded above by the Perron-Frobenius eigenvalue of a finite matrix, and this bound is attained if and only if the group is both amenable and unimodular.

Subharmonic functions on graphs

We study the behaviour of subharmonic functions on a graph. We assume bounds on the growth of balls and functions in order to obtain Liouville type theorems.

Some remarks on the weak maximum principle

We obtain a maximum principle at infinity for solutions of a class of nonlinear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumptions of volume growth conditions. In the case of the Laplace-Beltrami operator we relate our results to stochastic completeness and parabolicity of the manifold.

Brownian Motion and Harmonic Functions on Sol(p,q)

The Lie group Sol(p,q) is the semidirect product induced by the action of the real numbers R on the plane R^2 which is given by (x,y) --> (exp{p z} x, exp{-q z} y), where z is in R. Viewing Sol(p,q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift term in the z-variable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and compute the rate of escape. Also, we introduce the natural geometric compactification of Sol(p,q) and explain how Brownian motion converges almost surely to the boundary in the resulting topology. We also study all positive harmonic functions for the Laplacian with drift, and determine explicitly all minimal harmonic functions. All this is carried out with a strong emphasis on understanding and using the geometric features of Sol(p,q), and in particular the fact that it can be described as the horocyclic product of two hyperbolic planes with curvatures -p^2 and -q^2, respectively.

Liouville properties on graphs

We introduce a class of “differential operators” on graphs and we prove an energy estimate and a Liouville type theorem depending on some structural properties of the operators considered.

Tangential Boundary Behaviour of Harmonic Functions on Trees

We study the behaviour of harmonic functions on a homogeneous tree from the point of view of the tangential boundary covergence.

The Cost of Distinguishing Graphs

In a graph a set of vertices that is stabilized setwise by only the trivial automorphism is called a distinguishing class. Not every graph has such a set, but if it does, we call its minimum size the distinguishing cost. Many families of graphs have such sets, and for some families the distinguishing costs are surprisingly small. The talk begins with a survey of results about about the distinguishing cost for finite and infinite graphs. Then it concentrates on infinite graphs with finite cost and new bounds on the distinguishing cost of graphs with linear growth, two ends and either infinite automorphism group, or finite group with infinite motion. There remain interesting, unresolved problems.

A Proof of the Subadditive Ergodic Theorem

This is a presentation of the subadditive ergodic theorem. A proof is given that is an extension of F. Rieszs approach to the Birkhoff ergodic theorem.

Functions of Exponential Growth in a Half-Plane, Sets of Uniqueness, and the Müntz–Szász Problem for the Bergman Space

We introduce and study some new spaces of holomorphic functions on the right half-plane