Infinite Dimensional Analysis, Quantum Probability and Related Topics

We prove that the integral powers of the semicircular distribution are freely infinitely divisible. As a byproduct we get another proof of the free infinite divisibility of the classical Gaussian distribution.

It is proved in the paper that different natural Fourier transforms (FTs) of the measure mentioned in the title are not continuous with respect to such sufficient topologies as Sazonov and Gross–Sazonov (introduced by Smolyanov [Gross–Sazonov theorem for alternating cylindrical measures, Vestnik Moskov. Univ. (4) (1983) 4–12]) topologies. The motivation of the result is the fact that the FT of the standard Wiener measure is discontinuous in a known sufficient topologies as stated by Smolyanov and Fomin (see e.g., [Measures on Topological Linear Spaces, Uspekhi Mat. Nauk31(4) (1976) 3–56]).

We investigate the limit distributions of the discrete time quantum random walks on lattice spaces via a spectral analysis of concretely given self-adjoint operators. We discuss the interacting Fock spaces associated with the limit distributions. Thereby, we represent the moments of the limit distribution by vacuum expectation of the monomials of the Fock operator. We get formulas not only for one-dimensional walks but also for high-dimensional walks.

A notion of a nonlinear quantum dynamical semigroup is introduced and discussed. Some sufficient conditions, expressed solely in terms of the duality map, in order that a multivalued mapping on a C*-algebra generates the nonlinear Markov semigroup are proposed.

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

We figure out the intrinsic branching structure within (L-1) random walk in random environment. As applications, the branching structure enable us to calculate the expectation of the first hitting time directly, and specify the density of the invariant measure for the Markov chain of "the environment viewed from particles" explicitly.

Motivated by evolution equations on Clifford algebras and illustrated with the n-particle fermion algebra, a theory of invertible left- and right-Appell systems is developed for Clifford algebras of an arbitrary quadratic form. This work extends and clarifies the authors' earlier work on Clifford Appell systems, operator calculus, and operator homology/cohomology. A direct connection is also shown between blade factorization algorithms and the construction of Appell systems in these algebras.

Using the structure of the Boson-Fermion Fock space and an argument taken from [P. Bieliavsky, M. Cahen, S. Gutt, J. Rawnsley and L. Schwachhofer, Symplectic connections, Int. J. Geom. Meth. Mod. Phys.3 (2006) 375–420], we give a new proof of the triviality of the L² cohomology groups on an abstract Wiener space, alternative to that given by Shigekawa [De Rham–Hodge–Kodaira's decomposition on an abstract Wiener space, J. Math. Kyoto. Univ.26(2) (1986) 191–202]. We apply the representation theory of the symmetric group to characterize the spaces of exact and co-exact forms in their Boson-Fermion Fock space representation.

In this paper, motivated by Nica and Speicher [Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, Vol. 335 (Cambridge University Press, 2006)] and Kubo and Kuo [MRM-factors for the probability measures in the Meixner class, Infin. Dimens. Anal. Quantum Probab. Relat. Top.13 (2010) 525–550], we characterize a particular nilpotent case of a truncated forward shift operator by applying the technique of the random walks with repeated reflections and associated renewal equations. We also establish a stochastic order relationship by applying the crossing criteria.

For α, β ≥ 1. The space is the dual space of introduced by Duran in [Laguerre expansions of Gelfand–Shilov spaces, J. Approx. Theory74 (1993)]. We give a structure for the space , α ≥ 1 as follows: if and only if there exist a sequence {b_m} ⊂ (0, ∞) and a continuous bounded function f on (0, ∞) such that for every d > 0 there exists a constant C > 0 satisfying