Infinite Dimensional Analysis, Quantum Probability and Related Topics

The Radon transform is one of the most useful and applicable tools in functional analysis. First constructed by John Radon in 1917¹¹ it has now been adapted to several settings. One of the principle theorems involving the Radon transform is the Support Theorem. In this paper, we discuss how the Radon transform can be constructed in the white noise setting. We also develop a Support Theorem in this setting.

This paper is concerned with the notion of covariation for Banach space-valued processes. In particular, we introduce a notion of quadratic variation, which is a generalization of the classical restrictive formulation of Métivier and Pellaumail. Our approach is based on the notion of χ-covariation for processes with values in two Banach spaces B₁ and B₂, where χ is a suitable subspace of the dual of the projective tensor product of B₁ and B₂. We investigate some C¹ type transformations for various classes of stochastic processes admitting a χ-quadratic variation and related properties. If 𝕏¹ and 𝕏² admit a χ-covariation, Fⁱ : B_i → ℝ, i = 1, 2 are of class C¹ with some supplementary assumptions, then the covariation of the real processes F¹(𝕏¹) and F²(𝕏²) exist.

A detailed analysis is provided on the so-called window processes. Let X be a real continuous process; the C([-τ, 0])-valued process X(⋅) defined by X_t(y) = X_t+y, where y ∈ [-τ, 0], is called window process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. Those will constitute a significant Fukushima decomposition for functionals of windows of (weak) Dirichlet processes. As application, we provide a new technique for representing a path-dependent random variable as its expectation plus a stochastic integral with respect to the underlying process.

In this paper, we show the Basel problem via the beta distributions, which include the free Poisson distribution and positive arcsine law. This is a generalization of Ref. 2 by Fujita. We also obtain special values of an extension of generalized Hurwitz–Lerch zeta function, which was introduced by Gang, Jain and Kalla.

Geometric positions of square roots of coherent states of CCR algebras are investigated along with an explicit formula for transition amplitudes among them, which is a natural extension of our previous results on quasi-free states and will provide a new insight into the quasi-equivalence problem on coherent states.

We study two equivalent characterizations of the strong Feller property for a Markov process and of the associated sub-Markovian semigroup. One is described in terms of locally uniform absolute continuity, whereas the other uses local Orlicz-ultracontractivity. These criteria generalize many existing results on strong Feller continuity and seem to be more natural for Feller processes. By establishing the estimates of the first exit time from balls, we also investigate the continuity of harmonic functions for Feller processes which enjoy the strong Feller property.

Motivated by the Schwartz space of tempered distributions and the Kondratiev space of stochastic distributions we define a wide family of nuclear spaces which are increasing unions of (duals of) Hilbert spaces , with decreasing norms ‖⋅‖_p. The elements of these spaces are functions on a free commutative monoid. We characterize those rings in this family which satisfy an inequality of the form ‖f * g‖_p ≤ A(p - q)‖f‖_q‖g‖_p for all p ≥ q + d, where _* denotes the convolution in the monoid, A(p - q) is a strictly positive number and d is a fixed natural number (in this case we obtain commutative topological ℂ-algebras). Such an inequality holds in , but not in . We give an example of such a ring which contains . We characterize invertible elements in these rings and present applications to linear system theory.

We prove that the quadratic second quantization of an operator p on L²(ℝ^d) ∩ L^∞(ℝ^d) is an orthogonal projection on the quadratic Fock space if and only if p is a multiplication operator by a characteristic function χ_I, I ⊂ ℝ^d.

Let and be commuting unital C* subalgebras of , it is shown that and are C* independent if and only if