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 disjoint boundary arcs is required. Whenever $\lambda$ is in the resolvent of $P$ 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 obtain 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 $\lambda$-Martin kernel, there may be further choices for the edge weights which give rise to another $\lambda$-Poisson kernel with associated integral representations. In particular, we compare the resulting distributions on the boundary.

The material presented here is closely related to the contents of our “companion” paper\cite{PiWo}.

Chapter 3 studies semigroups on function spaces obtained via convolution semigroups of probability measures. Motivating examples that are studied in detail are the heat kernel (Brownian motion) and the Poisson kernel (Cauchy process). The characteristic functional (Fourier transform) is used to establish the Levy–Khinchine formula, and applications are given to stable laws. The generator and the semigroup are written as pseudo-differential operators.

We study the relationship between Hardy spaces of functions on the polytorus and certain spaces of holomorphic functions. We deal first with functions in finitely many variables, and later we jump to the infinite dimensional setting. For each N we consider the space of holomorphic functions g on the N-dimensional polydisc for which the L_p norms of g(rz) for 0<r<1 are bounded (known as the Hardy space of holomorphic functions). For each p these two Hardy spaces (of integrable functions on the N-dimensional polytorus and the N-dimensional polydisc) are isometrically isomorphic. The main tool in the proof is the Poisson operator (defined in Chapter 5). For the infinite dimensional case, we define the space of holomorphic functions g on l_2 ∩ Bc0 whose restrictions to the first N variables all belong to the corresponding Hardy space, and the norms are uniformly bounded (in N). These Hardy spaces of holomorphic functions on l_2 ∩ Bc0 and the Hardy spaces of integrable functions on the infinite-dimensional polytorus are isometrically isomorphic. The jump is given using a Hilbert criterion for Hardy spaces.

We work with integrable functions on the polytorus, both in finite and infinitely many variables. For such a function and a multi-index the corresponding Fourier coefficient is defined. For each 1 ≤ p ≤ ∞ the Hardy space H_p consists of those functions in L_p having non-zero Fourier coefficients only for multi-indices in the positive cone. The Hardy space H_\infty on the infinite dimensional polytorus and the space of bounded holomorphic functions on Bc0 are isometrically isomorphic. To prove this the Poisson kernel in several variables is defined, and the Poisson operator (defined through convolution with this kernel) is considered. With these it is shown that the trigonometric polynomials are dense in L_p for 1 ≤ p < ∞ and weak*-dense in L_\infty, and that so also are the analytic trigonometric polynomials in H_p and H_∞. The isometry between the two spaces is first established for the finite dimensional polytorus/polydisc and then, using a version of Hilbert’s criterion (see Chapter 2), raised to the infinite-dimensional case. The density of the polynomials can be proved using the Féjer kernel instead of the Poisson one.

Given a Banach space X, we consider Hardy spaces of X-valued functions on the infinite polytorus, Hardy spaces of X-valued Dirichlet series (defined as the image of the previous ones by the Bohr transform), and Hardy spaces of X-valued holomorphic functions on l_2 ∩ B_{c0}. The chapter is dedicated to study the interplay between these spaces. It is shown that the space of functions on the polytorus always forms a subspace of the one of holomorphic functions, and these two are isometrically isomorphic if and only if X has ARNP. Then the question arises of what do we find in the side of Dirichlet series when we look at the image of the Hardy space of holomorphic functions. This is also answered, showing that this consists of Dirichlet series for which all horizontal translations (those whose coefficients are (a_n/n^ε)) are in \mathcal{H}_p with uniformly bounded norms. Also, a version of the brothers Riesz theorem for vector-valued functions is given.