Infinite Dimensional Analysis, Quantum Probability and Related Topics

Let {α(n); n≥0} be a sequence of positive numbers satisfying certain conditions. A Gel'fand triple associated with the sequence {α (n); n ≥ 0} has been introduced on a white noise space (ℰ′,μ) by Cochran, Kuo and Sengupta. In this paper we obtain additional conditions on the sequence {α(n); n≥0} in order to carry out white noise distribution theory on the space (ℰ′,μ). Moreover, we show that the Bell numbers satisfy these additional conditions.

In this paper, we study lower order perturbations of a symmetric second-order differential operator generating a hypercontractive semigroup. We give a probabilistic representation of the, in general, not sub-Markovian semigroup associated with the perturbed operator and prove that the perturbed semigroup is also hypercontractive under some exponential integrability conditions on the coefficients.

A U(N)-invariant matrix model with d matrix variables is studied. It was shown that in the limit N → ∞ and d→0 the model describes the knot diagrams. We realize the free partition function of the matrix model as the generalized expectation of a Hida distribution Φ_N,d. This enables us to give a mathematically rigorous meaning to the partition function with interaction. For the generalized function Φ_N,d, we prove a Wick theorem and we derive explicit formulas for the propagators.

We extend the Gross derivative to a space of generalized random variables which have a (formal) chaos expansion with kernels from the space of tempered Schwartz distributions. The extended derivative, which we call the Hida derivative, has to be interpreted in the sense of distributions. Many of the properties of the Gross derivative are proved to hold for the extension as well. In addition, we derive a representation formula for the Hida derivative involving the Wick product and a centered Gaussian random variable. We apply our results to calculate the Hida derivative of a class of stochastic differential equations of Wick type.

A classical stochastic process which is Markovian for its past filtration is also Markovian for its future filtration. We show with a counterexample based on quantum liftings of a finite state classical Markov chain that this property cannot hold in the category of expected Markov processes. Using a duality theory for von Neumann algebras with weights, developed by Petz on the basis of previous results by Groh and Kümmerer, we show that a quantum version of this symmetry can be established in the category of weak Markov processes in the sense of Bhat and Parthasarathy. Here time reversal is implemented by an anti-unitary operator and a weak Markov process is time reversal invariant if and only if the associated semigroup coincides with its Petz dual. This construction allows one to extend to the quantum case, both for backward and forward processes, the Misra–Prigogine–Courbage internal time operator and to show that the two operators are intertwined by the time reversal anti-automorphism.

We prove tightness of capacities generated by Sobolev classes of all orders in locally convex spaces under quite mild conditions on the space and on the measure.

For certain unitary representations of a Lie algebra we define the statistical manifold ℳ of states as the convex cone of for which the partition function Z=Trexp{-X} is finite. The Hessian of Ψ=log Z defines a Riemannian metric g on , (the Bogoliubov–Kubo–Mori metric); foliates into the union of coadjoint orbits, each of which can be given a complex structure (that of Kostant). The program is carried out for so(3), and for sl(2,R) in the discrete series. We show that ℳ=R⁺×CP¹ and R⁺×H respectively. We show that for the metaplectic representation of the quadratic canonical algebra, ℳ=R⁺×CP²/Z₂. Exactly solvable model dynamics is constructed in each case.

Let γ be a Radon Gaussian measure on a locally convex space X with the Cameron–Martin space H, let A⊂X be a γ-measurable set, and let F: A→ E be a γ-measurable mapping with values in a separable Hilbert space E such that |F(x)-F(y)|_E ≤C|x-y|_H whenever x, y∈A, x-y∈H. The main result in this work gives a γ-measurable extension of F to all of X such that |F(x+h)-F(x)|_E≤C|h|_H for all x∈X and h∈H. Some related results are obtained.