tt-Martin boundary of killed random walks in the quadrant

We compute the $t$-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete $t$-harmonic functions. Our approach is uniform in $t$, and shows that there are three regimes for the Martin boundary.Comment: 18 pages, 2 figures, to appear in S\'eminaire de Probabilit\'e

On twisted Fourier analysis and convergence of Fourier series on discrete groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.Comment: 35 pages; abridged, revised and update

Heat Diffusion on Homogeneous Trees

AbstractLet X be a homogeneous tree. We study the heat diffusion process associated with the nearest neighbour isotropic Markov operator on X. In particular it is shown that the heat maximal operator is weak type (1, 1) and strong type (p, p), for every 1 <p < ∞. We estimate the asymptotic behaviour of the heat maximal function. Moreover, we introduce a family of Hp spaces on X. It is proved that Hp=lp(X) for 1 <p < ∞ and is conjectured that Hp for p less than 1, is trivial

The Algebras Generated by the Laplace Operators in a Semi-homogeneous Tree

In a semi-homogeneous tree, the set of edges is a transitive homogeneous space of the group of automorphisms, but the set of vertices is not (unless the tree is homogeneous): in fact, the latter splits into two disjoint homogeneous spaces V+, V− according to the homogeneity degree. With the goal of constructing maximal abelian convolution algebras, we consider two different algebras of radial functions on semi-homogeneous trees. The first consists of functions on the vertices of the tree: in this case the group of automorphisms gives rise to a convolution product only on V+ and V− separately, and we show that the functions on V+, V− that are radial with respect to the natural distance form maximal abelian algebras, generated by the respective Laplace operators. The second algebra consists of functions on the edges of the tree: in this case, by choosing a reference edge, we show that no algebra that contains an element supported on the disc of radius one is radial, not even in a generalized sense that takes orientation into account. In particular, the two Laplace operators on the edges of a semi-homogeneous (non-homogeneous) tree do not generate a radial algebra, and neither does any weighted combination of them. It is also worth observing that the convolution for functions on edges has some unexpected properties: for instance, it does not preserve the parity of the distance, and the two Laplace operators never commute, not even on homogeneous trees

Spherical functions and harmonic analysis on free groups

Let F,, r &gt; 1, be a free group with r generators. In this paper, we study a principal series and a complementary series of irreducible unitary representations of F,, which are defined through the action of F, on its Poisson boundary, relative to a simple random walk. We show that the regular representation of F, can be written as a direct integral of the representations of the principal series and that the resulting harmonic analysis on the free group bears a close resemblance with the harmonic analysis of SL(2, IR)

Multiple boundary representations of λ-harmonic functions on trees

We consider a countable tree $T$, possibly having vertices with infinite degree, and an arbitrary stochastic nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with eigenvalue $lambda in C$, under the condition that the oriented edges can be equipped with complex-valued weights satisfying three natural axioms. These axioms guarantee that one can construct a $lambda$-Poisson kernel. The boundary integral is with respect to distributions, that is, elements in the dual of the space of locally constant functions. Distributions are interpreted as finitely additive complex measures. In general, they do not extend to $sigma$-additive measures: for this extension, %a summability condition over the neighbours of a vertex is required. a summability condition over disjoint boundary arcs is required. as a self-adjoint operator on a naturally associated $ell^2$-space and the diagonal elements of the resolvent (``Green function'') do not vanish at $lambda$, one can use the ordinary edge weights corresponding to the Green function, and one gets the ordinary $la$-Martin kernel. We then consider the case when $P$ is invariant under a transitive group action. In this situation, we study the phenomenon that in addition to the $la$-Martin kernel, there may be further choices for the edge weights which give rise to another $la$-Poisson kernel with associated integral representations. In particular, we compare the resulting distributions on the boundary