Infinite Dimensional Analysis, Quantum Probability and Related Topics

A classical deterministic, reversible dynamical system, reproducing the Einstein–Podolsky–Rosen (EPR) correlations in full respect of causality and locality and without the introduction of any ad hoc selection procedure, was constructed in Ref. 4.

In this paper we prove that the above-mentioned model is unique (see Theorem 3.1) in the sense that any local causal probability measure which reproduces the EPR correlations must coincide, under natural and generic assumptions, with the one constructed in Ref. 4.

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

The bm-central limit theorem for positive-definite real symmetric matrices and for homogeneous rooted trees are studied. The combinatorial nature of it is described. In addition, the construction of bm-product of graphs is given, which generalizes comb and boolean products. General properties of bm-extension operators are shown.

We derive a large deviation principle for occupation time of super α-stable process in ℝ^d with d > 2α. The decay of tail probabilities is shown to be exponential and the rate function is characterized. Our result can be considered as a counterpart of Lee's work on large deviations for occupation times of super-Brownian motion in ℝ^d for dimension d > 4 (see Ref. 10).

Motivated by physical problems, Misra, Prigogine and Courbage (MPC) studied the following problem: given a one-parameter unitary group {U_t} on a separable Hilbert space , find a Hilbert space , a contraction semigroup {W_t} on and an injective operator with dense range which intertwines the actions of {U_t} and {W_t} (ΛW_t = U_t Λ). More precisely, they studied the case where is an L²-space over a probability space and both {U_t} and {W_t} are Markovian (i.e. positivity and identity preserving). MPC gave a sufficient condition for the existence of a solution of the above problem, the existence of a time operator associated to {U_t}. In this paper we prove that, using the Sz.-Nagy–Foiaş dilation theory, it is possible to give a constructive characterization of all the solutions of the MPC problem in the general context. This criterium allows one to construct a solution of the MPC problem for which no time operator exists. When specialized to L²-spaces and Markovian {U_t} and {W_t}, the present criterium is applied to address the so-called inverse problem of Statistical Mechanics, namely to characterize the intrinsically random dynamics {U_t}.

In this paper, we shall give the representation of a semicircular perturbation of the free Fisher information by the mean of the conditional variance , where S is a standard semicircular element freely independent of X, and ε > 0. Using this representation, we will give alternative proofs of the free Fisher information inequality, in which the free analogue of Stam's inequality can be obtained as a special case, and of the free entropy power inequality in an infinitesimal approach to the free entropy.

Gaussian and Poisson noises can be discussed not only in parallel, but we can also see dissimilarity between them together with some interesting results by using the space of quadratic Hida distributions. Further we can discuss calculus of their functionals to see a duality of the two noises.

Let 𝔐 be the Banach space of σ-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of 𝔐 is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon–Nikodým theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen–Wintner purity theorem for our decompositions.

In Ref. 9, Kosaki proved an uncertainty principle for matrices, related to Wigner–Yanase–Dyson information, and asked if a similar inequality could be proved in the von Neumann algebra setting. In this paper we prove such an uncertainty principle in the semifinite case.