Infinite Dimensional Analysis, Quantum Probability and Related Topics

In this paper we study a class of quantum Markov semigroups whose restriction to an abelian sub-algebra coincides, on the configurations with finite support, with the exclusion type semigroups introduced in Liggett's book¹⁴ of exchange rates not symmetric in the index site r, s. We find a sufficient condition for the existence of infinitely many faithful diagonal (or classical) invariant states for the semigroup, that satisfy a quantum detailed balance condition. This class of semigroups arises naturally in the stochastic limit of quantum interacting particles in the sense of Accardi and Kozyrev.¹ We call these semigroups asymmetric exclusion quantum Markov semigroups and the associated processes asymmetric exclusion quantum processes.

We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line.

In this paper, it will be shown that a probability measure on ℂ associated with the Jacobi–Szegö parameters of the orthogonal polynomials can be obtained by making use of the classical Mellin transform and its convolution property. We shall construct several measures on ℂ represented by the modified Bessel functions. The material in this paper gives nontrivial examples originated from the continuous dual Hahn polynomials (one of hypergeometric orthogonal polynomials), which are beyond the Meixner–Pollaczek polynomials appeared in our previous papers.^{4, 5}

We prove global well-posedness in the strong sense for stochastic generalized porous media equations driven by a square integrable martingale with stationary independent increments.

In this paper, we study the mapping r ↦ δ_r(Q), where Q is an observable satisfying mild conditions and δ_r(Q) is the delta function of Q with r ∈ ℝ. We find that, under certain integrable conditions, the mapping r ↦ δ_r(Q) is just the Radon–Nikodym derivative of Q's spectral measure with respect to Lebesgue measure and, moreover, Q can be represented as an integral of the mapping r ↦ rδ_r(Q) with respect to Lebesgue measure. An example is also given.

Building on the ideas of Ref.¹⁴ we indicate how the concept of a composition operator may be extended to the context of Haagerup L^p-spaces.

This note is devoted to extension of quantum probability calculus to generalizations of complex Hilbert space. Starting with Hilbert space over complex hyperbolic numbers, we derive general hyper-trigonometric interference of probabilities.

An explicit formula is found for the antipode in Itô–Hopf algebras. The Hopf algebra structure is extended to iterated classical and quantum stochastic integrals. The antipode is related to change of orientation in the simplicial domains of such integrals.

We consider the renormalized square of the solution of the stochastic heat equation and obtain for this process a new dynamic involving the second quantization of an unbounded operator. This is achieved by a version of the Itô formula which we derive by a simple application of the Wick chain rule.

We show that a continuous action of a quantum semigroup on a finite quantum space (finite dimensional C^*-algebra) preserving a faithful state comes from a continuous action of the quantum Bohr compactification of . Using the classification of continuous compact quantum group actions on M₂, we give a complete description of all continuous quantum semigroup actions on this quantum space preserving a faithful state.

Let E → M be a vector bundle over a compact manifold. In this short note we prove that the Schrödinger semigroup (complex heat semigroup) associated to a self-adjoint (non-negative self-adjoint) operator in Γ_L²(M, E) corresponds to a Γ_L²(M, E)^*-valued Hida distribution, which depends continuously (holomorphically) on the time parameter.