Infinite Dimensional Analysis, Quantum Probability and Related Topics

In this paper we shall discuss the Lévy Laplacian acting on the space of Poisson noise functionals. We give eigenfunctions of the Laplacian and extend the operator to a self-adjoint operator densely defined on some Hilbert space.

In this paper, we are mainly concerned with the computation of the dynamical entropy for the noncommutative toral automorphisms and the noncommutative shifts on a spin chain. The entropy that we consider is based on coarse-grained models of the system. In contrast to the classical situation, successive observations of a quantum system can generate entropy by themselves. We pay special attention to the delicate problem of delimiting a natural class of models for computing the entropy.

This is the first part of a work on Laplace method for the asymptotics of integrals with respect to smooth measures and a large parameter developed in infinite dimensions. Here the case of finitely many (nondegenerate) minimum points is studied in details. Applications to large parameters behavior of expectations with respect to probability measures occurring in the study of systems of statistical mechanics and quantum field theory are mentioned.

We present a complete description of the extreme points of the convex set of all decision rules in testing multiple hypotheses concerning a quantum system whose states are described by density matrices in a finite-dimensional Hilbert space. Holevo's equations for an optimal decision are derived by methods of elementary calculus and a simple example is given in order to illustrate the nonuniqueness of optimal decision rules.

Existence and continuity results are obtained for self-intersection local time of -valued Ornstein–Uhlenbeck processes , where X₀ is Gaussian, W_t is an -Wiener process (independent of X₀), and T'_t is the adjoint of a semigroup T_t on . Two types of covariance kernels for X₀ and for W are considered: square tempered kernels and homogeneous random field kernels. The case where T_t corresponds to the spherically symmetric α-stable process in ℝ^d, α∈(0,2], is treated in detail. The method consists in proving first results for self-intersection local times of the ingredient processes: W_t, T'_t X₀ and , from which the results for X_t are derived. As a by-product, a class of non-finite tempered measures on ℝ^d whose Fourier transforms are functions is identified. The tools are mostly analytical.

We calculate the probability measures for some linear combinations of a free family of projections by using Voiculescu's R-transform for free additive convolution in the free probability theory. The measures we give in this paper include many examples that have been calculated, for instance, the spectral measure for the adjacency operators of the infinite distance regular graphs, and the Plancherel measures associated with free product groups. Furthermore, we find the recursive relations for the orthogonal polynomials associated with them.

We characterize a class of quantum Markov states in terms of a locality property of their modular automorphism group or, equivalently, of their φ-conditional expectations and we give an explicit description of the structure of these states. This study is meant as a starting point for the investigation of the structure of Markovian KMS-states of quantum spin chains as well as of multidimensional quantum spin lattices.