Infinite Dimensional Analysis, Quantum Probability and Related Topics

In this paper, we shall construct probability measures on ℂ, which can be expressed by the Mellin convolution product of the modified Bessel functions. Our measures on ℂ are related to the Jacobi–Szegö parameters for the continuous dual Hahn polynomials S_n(x², a, b, c) under the special choice of parameters a, b, c. This paper contains new materials which go further than our previous results in Ref.7. The most interesting thing is that the Mellin convolution of two modified Bessel functions can again be expressed in terms of the modified Bessel functions, by choosing parameters a, b, c appropriately. The origin of our research in this direction goes back to the Bargmann–Fock representation of the classical non-Gaussian random variables.^5–8 Our results would have a potential to be related to the higher powers of creation and annihilation operators acting on a new class of interacting Fock spaces.

An important connection between the finite-dimensional Gaussian Wick products and Lebesgue convolution products will be proven first. Then this connection will be used to prove an important Hölder inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the Hölder and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian Hölder inequality and classic Hölder inequality, between the Nelson hypercontractivity and classic Young inequality with the sharp constant, and between the third more general inequality and an extension by Lieb of the Young inequality with the best constant. Since the Gaussian probability measure exists even in the infinite-dimensional case, the above three inequalities can be extended, via a classic Fatou's lemma argument, to the infinite-dimensional framework.

We construct a growing graph such that the Jacobi coefficient of the asymptotic spectral distribution is . The distribution is a χ² distribution with one degree of freedom normalized to have mean 0 and variance 1.

Starting with a complex Hilbert space, using inductive limits, we build Lie algebras, and find families of representations. They include those often studied in mathematical physics in order to model quantum statistical mechanics or quantum fields. We explore natural actions on infinite tensor algebras T(H) built with a functorial construction, starting with a fixed Hilbert space H.

While our construction applies also when H is infinite-dimensional, the case with N ≔ dim H finite is of special interest as the symmetry group we consider is then a copy of the non-compact Lie group U(N, 1). We give the tensor algebra T(H) the structure of a Hilbert space, i.e. the unrestricted infinite tensor product Fock space . The tensor algebra T(H) is naturally represented as acting by bounded operators on , and U(N, 1) as acting as a unitary representation. From this we built a covariant system, and we explore how the fermion, the boson, and the q on Hilbert spaces are reduced by the representations. In particular we display the decomposition into irreducible representations of the naturally defined U(N, 1) representation.

In this paper we study forward quantum Markov chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and define QMC as a weak limit of those states which depends on the boundary conditions. Using the provided construction, we investigate QMC associated with XY-model on a Cayley tree of order two. We prove uniqueness of QMC associated with such a model, this means the QMC does not depend on the boundary conditions.

We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in noncommutative probability theory and unifies the monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the additive convolution and deformations of the monotone convolution. We define cumulants for a general convolution to analyze the deformed convolutions.

In this paper, we construct the solutions of semilinear parabolic PDEs with singular coefficients and establish the link to solutions of backward stochastic differential equations.