Infinite Dimensional Analysis, Quantum Probability and Related Topics

In this paper, we present the Bogoliubov’s causal perturbative QFT, which includes only one refinement: the creation-annihilation operators at a point, i.e. for a specific momentum, are mathematically interpreted as the Hida operators from the white noise analysis. We leave the rest of the theory completely unchanged. This allows avoiding infrared — and ultraviolet — divergences in the transition to the adiabatic limit for interacting fields. We present here the analysis of the causal axioms for the scattering operator with the Hida operators as the creation-annihilation operators.

In a recent paper, Lanconelli and Scorolli extended to the multidimensional case a Wong–Zakai-type approximation for Itô stochastic differential equations (SDEs) proposed by Øksendal and Hu. The aim of this paper is to extend the latter result to system of SDEs of Itô type driven by fractional Brownian motion (fBm) like those considered by Hu. This extension is not trivial since the covariance structure of the fBm precludes us from using the same approach as that used by Lanconelli and Scorolli. Instead we employ a truncated Cameron–Martin expansion as the approximation for the fBm. We are naturally led to the investigation of a semilinear hyperbolic system of evolution equations in several space variables that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the number of terms in the expansion goes to infinite.

In this paper, we prove the existence of a mild $L^p$-solution for the backward stochastic evolution inclusion (BSEI for short) of the form

The case when the processes with values in spaces that have martingale type $2$ has been also studied.

In this paper, we consider a Hard-Core (HC) model with two spin values on Cayley trees. The conception of alternative Gibbs measure is introduced and translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant. In addition, we study free energy of the model.

In this paper, we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure ${\pi }_{\sigma }^{\beta }$, $0<\beta \le 1$, on the dual of Schwartz test function space ${{𝒟}}^{\prime }$. The Hilbert space $L^2({\pi }_{\sigma }^{\beta })$ of complex-valued functions is described in terms of a system of generalized Appell polynomials ${\mathbb{P}}^{\sigma ,\beta ,\alpha }$ associated to the measure ${\pi }_{\sigma }^{\beta }$. The kernels $C_n^{\sigma ,\beta }(\cdot )$, $n\in {\mathbb{N}}_0$, of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system ${\mathbb{P}}^{\sigma ,\beta ,\alpha }$, there is a generalized dual Appell system ${\mathbb{Q}}^{\sigma ,\beta ,\alpha }$ that is biorthogonal to ${\mathbb{P}}^{\sigma ,\beta ,\alpha }$. The test and generalized function spaces associated to the measure ${\pi }_{\sigma }^{\beta }$ are completely characterized using an integral transform as entire functions.

In this paper, we study Gabor frames in the matrix-valued signal space $L^2(G,{ℂ}^{n\times n})$, where $G$ is a locally compact abelian group which is metrizable and $\sigma$-compact, and $n$ is a positive integer. First, we give sufficient conditions on scalars in an infinite combination of vectors (from a given matrix-valued Gabor frame) to constitute a new frame for the space $L^2(G,{ℂ}^{n\times n})$. This generalizes a result due to Aldroubi. Second, we discuss frame conditions for finite sums of matrix-valued Gabor frames. Sufficient conditions for finite sums of matrix-valued Gabor frames in terms of frame bounds are established. It is shown that the sum of images of matrix-valued Gabor frames under bounded linear operators acting on $L^2(G,{ℂ}^{n\times n})$ constitute a frame for the space $L^2(G,{ℂ}^{n\times n})$ provided operators are adjointable with respect to the matrix-valued inner product and satisfy a majorization. Finally, we show that matrix-valued Gabor frames are stable under small perturbations.

Motivated by measuring the degree of similarity of a pair of quantum states (density matrices), we consider the metric property of the modified Bures angles and modified Bures distances of symmetric functions which are extensions of some fidelity measures on the spaces ${𝒫}$ of nonzero positive semi-definite matrices. We use the positive semi-definiteness of the Gram-type matrices to characterize the metric property of the modified Bures angles. As a consequence, we can show that the modified Bures angles induced by the geometric mean, harmonic mean, minimum and maximum of two positive numbers are metrics on ${𝒫}$. In addition, we can also show that the metric property of the modified Bures angles is stronger than that of the modified Bures distances.