Infinite Dimensional Analysis, Quantum Probability and Related Topics

An incompressible ideal fluid in the two-dimensional torus (i.e. the Euler equation in a rectangle with periodic boundary conditions) is considered. The flow for a vorticity field concentrated in any finite number of points is analyzed. A compound Poisson measure Π, invariant for this flow, is introduced. The Hilbert space ℒ²(Π) and the properties of the corresponding ℒ²-flow are investigated. In particular it is proven that the corresponding generator is Markov unique.

We revisit a CKS-space from the viewpoint of standard setup of white noise calculus and prove a general characterization theorem for white noise operators from a CKS-space into another CKS-space. The usual characterizations so far obtained for white noise distributions, white noise test functions and white noise operators are all consequences of the unified characterization theorem.

Feynman's path integral in adelic quantum mechanics is considered. The propagator for one-dimensional adelic systems with quadratic Lagrangians is analytically evaluated. Obtained exact general formula has the form which is invariant under interchange of the number fields ℝ and ℚ_p.

The renormalized stochastic differentials of the square of white noise are defined in Boson–Fock space representation. The Itô multiplication table of these differentials and the module form of the unitarity conditions are obtained.

In this article we study the long time behaviour of a class of stochastic differential equations introduced in the theory of measurements continuous in time for quantum open systems. Such equations give the time evolution of the a posteriori states for a system underlying a continual measurement. First of all we give conditions for the equation to preserve pure states and then, in the case of a finite-dimensional Hilbert space, we obtain sufficient conditions from which the stochastic equation for a posteriori states is ensured to map, for t → +∞, mixed states into pure ones. Finally we study existence and uniqueness of an invariant measure for the equations which preserve pure states. We give a general theorem for the purely diffusive case, again in the finite-dimensional case; we then apply it to some physical examples. For the purely jump case, an example is discussed in which the invariant measure exists and is unique.

Working within the framework of Hopf algebras, a random walk and the associated diffusion equation are constructed on a space that is algebraically described as the merging of the real line algebra with the anyonic line algebra. Technically this merged structure is a smash algebra, namely an algebra resulting from a braided tensoring of real with anyonic line algebras. The motivation of introducing the smashing results from the necessity of having noncommuting increments in the random walk. Based on the observable-state duality provided by the underlying Hopf structure, the construction is cast into two dual forms: one using functionals determined by density probability functions and the other using the associated Markov transition operator. The ensuing diffusion equation is shown to possess triangular matrix realization. The study is completed by the incorporation of Hamiltonian dynamics in the above random walk model, and by the construction of the dynamical equation obeyed by statistical moments of the problem for generic entangled density functions.

We study spectral gaps w.r.t. marginals of pinned Wiener measures on spaces of discrete loops (or, more generally, pinned paths) on a compact Riemannian manifold M. The asymptotic behaviour of the spectral gap as the time parameter T of the underlying Brownian bridge goes to 0 is investigated. It turns out that depending on the choice of a Riemannian metric on the base manifold, very different asymptotic behaviours can occur. For example, on discrete loop spaces over sufficiently round ellipsoids the gap grows of order α/T as T ↓ 0. The strictly positive rate α stabilizes as the discretization approaches the continuum limit. On the other extreme, if there exists a closed geodesic γ : S¹ → M such that the sectional curvature on γ(S¹) is strictly negative, and the loop is pinned close to γ(S¹), then the gap decays of order exp(-β/T), and the decay rate β approaches +∞ as the discretization approaches the continuum limit.

The quantum capacity of a pure quantum channel and that of classical-quantum-classical channel are discussed in detail based on the fully quantum mechanical mutual entropy. It is proved that the quantum capacity generalizes the so-called Holevo bound.

A full proof of the Feynman–Kac-type formula for heat equation on a compact Riemannian manifold is obtained using some ideas originating from the papers of Smolyanov, Truman, Weizsaecker and Wittich.^1-3 In particular, the technique exploited in the paper has some common lines with Chernoff theorem, which is one of the basic points of the approach to the topics undertaken in the above-mentioned papers.

In Wang,⁸ a class of interacting measure-valued branching diffusions with singular coefficient were constructed and characterized as a unique solution to ℒ^ε-martingale problem by a limiting duality method since in this case the dual process does not exist. In this paper, we prove that for any ε ≠ 0 the superprocess with singular motion coefficient is just the super-Brownian motion. The singular motion coefficient is handled as a sequential limit motivated by Antosik et al.¹ Thus, the limiting superprocess is investigated and identified as the motion coefficient converges to a singular function. The representation of the singular spacetime Itô's integral is derived.