Infinite Dimensional Analysis, Quantum Probability and Related Topics

We study positivity and contractivity properties for semigroups on , compute the optimal log-Sobolev constant and prove hypercontractivity for the class of positive semigroups leaving invariant both subspaces generated by the Pauli matrices σ₀, σ₃ and σ₁, σ₂. The optimal log-Sobolev constant turns out to be bigger than the usual one arising in several commutative and noncommutative contexts when the semigroup acts on the off-diagonal matrices faster than on diagonal matrices. These results are applied to the semigroup of the Wigner–Weisskopf atom.

In the hierarchy of freeness construction, free independence was reduced to tensor independence in the weak sense of convergence of moments. In this paper we show how to reduce free independence to tensor independence in the strong sense. We construct a suitable unital *-algebra of closed operators "affiliated" with a given unital *-algebra and call the associated closure "monotone". Then we prove that monotone closed operators of the form

We give a stochastic analog of the construction of McLaughlin of the string bundle on the loop space by using tools of algebraic topology.

We provide simple criteria for nonconservativity of a minimal quantum dynamical semigroup.

For random fields with fractional regularity order (respectively, fractional singularity order), an orthogonal decomposition of the associated reproducing kernel Hilbert space with respect to domains with fractal boundary is derived. The approach presented is based on the theory of generalized random fields on fractional Sobolev spaces. The orthogonal decomposition derived is equivalent to the weak-sense Markov condition, in the second-order moment sense, studied in Ref. 50, and based on the concept of splitting Hilbert spaces. A mean-square fractional order differential representation on bounded domains with fractal boundary is also obtained. In the Gaussian case, the random fields studied have fractal sample paths (see Ref. 1). Examples of fractional-order differential models in the class considered are provided.

The adjacency matrix of a comb graph is decomposed into a sum of monotone independent random variables with respect to the vacuum state. The vacuum spectral distribution is shown to be asymptotically the arcsine law as a consequence of the monotone central limit theorem. As an example the comb lattice is studied with explicit calculation.

We determine a new explicit representation of strong solutions of Itô-diffusions and elicit its correspondence to the general stochastic transport equation. We apply this formula to deduce an explicit Donsker delta function of a diffusion.

We introduce a notion of "Λ-boolean independence", which interpolates between the tensor and the boolean one. The central and the Poisson limit theorem are given in terms of the related orthogonal polynomials.

It has been shown in Refs. 2–6 that two natural definitions of surface measures, on the space of continuous paths in a compact Riemannian manifold embedded into ℝⁿ, introduced in the paper by Smolyanov¹ are equivalent; this means that there exists a natural object — the surface measure, which we call the Smolyanov surface measure. Moreover, it has been shown^2–6 that this surface measure is equivalent to the Wiener measure and the corresponding density has been found. But the known proof of the equivalence of the two definitions of the surface measure is rather nonexplicit; in fact the densities of the measures corresponding to the two different definitions were found independently and only a posteriori it was discovered that those densities coincided.

Our aim is to give a direct proof of this fact. We introduce a more restrictive definition of the surface measure as the weak limit of a standard Brownian motion in ℝⁿ conditioned to be in the tubular ε-neighborhood of the manifold at times 0=t₀<t₁<⋯<t_n-1<t_n= 1 as both ε and the diameter of the partition tend to zero. Letting ε and then the diameter of the partition tend to zero and vice versa, we arrive at the two definitions above. We prove the existence of the Smolyanov surface measure using our definition, show that this measure is equivalent to the law of a Brownian motion on the manifold, and compute the corresponding density in terms of the curvature of the manifold. As a special case of this, we again obtain the results of Refs. 2–6.

Using the weak coupling limit as a quantum functional central limit (in the sense of Accardi, Frigerio and Lu), we give a quantum Markovian description of the open XY-model, followed by the large-N limit. The treatment is noteworthy for the fact that the harmonic decomposition of the stochastic dynamics is not that of the free evolution, but rather arises from the macroscopic variables being studied. We conclude with a simple Hilbert module formulation of the dynamics.