Infinite Dimensional Analysis, Quantum Probability and Related Topics

It is known that the gamma distribution γ_κ is MRM-applicable for h(x) = e^x and for some hypergeometric functions also. We are interested in the problem to determine all possible MRM-factors of probability measures which are MRM-applicable for e^x. We may say that the measures are in Meixner class. Such typical measures are Gaussian, Poisson, gamma, negative binomial and Meixner distributions and others are obtained from their modifications by affine transforms. We will give the complete list of MRM-factors different from e^x up to trivial deformation:

(1) for gamma distribution γ_κ.

(2) for gamma distribution γ_κ.

(3) for standard Gaussian distribution.

(4) for shifted negative binomial distribution.

σ_βNegBin(κ,p) with κ = 2, β = 1, for Meixner distribution M_κ,η with κ = 2 and for gamma distribution γ_κ with κ = 2, which is a special case of (2) with c = 1.

In this paper we introduce the quadratic Weyl operators canonically associated to the one mode renormalized square of white noise (RSWN) algebra as unitary operator acting on the one mode interacting Fock space {Γ, {ω_n, n ∈ ℕ}, Φ} where {ω_n, n ∈ ℕ} is the principal Jacobi sequence of the nonstandard (i.e. neither Gaussian nor Poisson) Meixner classes.

We deduce the quadratic Weyl relations and construct the quadratic analogue of the Heisenberg Lie group with one degree of freedom. In particular, we determine the manifold structure of the group and introduce a local chart containing the identity on which the group law has a simple rational expression in the chart coordinates (see Theorem 6.3).

An intrinsic multi-type branching structure within the transient (1, R)-RWRE is revealed. The branching structure enables us to specify the density of the absolutely continuous invariant measure for the Markov chain of environments seen from the particle and reprove the LLN with a drift explicitly in terms of the environment.

The energy representation of a gauge group on a Riemannian manifold has been discussed by several authors. Y. Shimada has shown the irreducibility with the use of white noise analysis for compact Riemannian manifolds. In this paper we extend its technique to the noncompact Riemannian manifolds which have differential operators satisfying some conditions.

We construct noncommutative Brownian motions indexed by partially ordered subsets of Euclidean spaces. The noncommutative independence under consideration is the bm-independence and the time parameter is taken from a positive cone in a vector space (, the Lorentz cone or the positive definite real symmetric matrices). The construction extends the Muraki's idea of monotonic Brownian motion. We show that our Brownian motions have bm-independent increments for bm-ordered intervals. The appropriate version of the Donsker Invariance Principle is also proved for each positive cone. It requires the bm-Central Limit Theorems related to intervals in the given partially ordered set of indices.

In this paper, we derive a formula for the expectation of random Hermite polynomials in Skorohod integrals, extending classical results in the adapted case. As an application we recover, under simple conditions and with short proofs, the anticipative Girsanov identity and quasi-invariance results obtained in Ref. 6 for quasi-nilpotent shifts on the Wiener space.

We study the relationship between the type of the random walk on some random trees and the structure of those trees in terms of fractal and resistance dimensions. This paper generalizes some results of Refs. 8–10.