Infinite Dimensional Analysis, Quantum Probability and Related Topics

We prove that, each probability meassure on $ℝ$, with all moments, is canonically associated with (i) a ∗-Lie algebra; (ii) a complexity index labeled by pairs of natural integers. The measures with complexity index $(0,K)$ consist of two disjoint classes: that of all measures with finite support and the semi-circle-arcsine class (the discussion in Sec. 4.1 motivates this name). The class $C(\mu )=(0,0)$ coincides with the $\delta$-measures in the finite support case and includes the semi-circle laws in the infinite support case. In the infinite support case, the class $C(\mu )=(0,1)$ includes the arcsine laws, and the class $C(\mu )=(0,2)$ appeared in central limit theorems of quantum random walks in the sense of Konno. The classes $C(\mu )=(0,K)$, with $K\ge 3$, do not seem to be present in the literature. The class $(1,0)$ includes the Gaussian and Poisson measures and the associated ∗-Lie algebra is the Heisenberg algebra. The class $(2,0)$ includes the non-standard (i.e. neither Gaussian nor Poisson) Meixner distributions and the associated ∗-Lie algebra is a central extension of $\mathrm{\text{sl}}(2,ℝ)$. Starting from $n=3$, the ∗-Lie algebra associated to the class $(n,0)$ is infinite dimensional and the corresponding classes include the higher powers of the standard Gaussian.

We study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida–Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the corresponding Hamiltonians. This is used to study applications to insider games in finance, specifically optimal insider consumption and optimal insider portfolio under model uncertainty.

We calculate the distribution with respect to the vacuum state of the distance-$k$ graph of a $d$-regular tree. From this result we show that the distance-$k$ graph of a $d$-regular graphs converges to the distribution of the distance-$k$ graph of a regular tree. Finally, we prove that, properly normalized, the asymptotic distributions of distance-$k$ graphs of the $d$-fold free product graph, as $d$ tends to infinity, is given by the distribution of $P_k(s)$, where $s$ is a semicirlce random variable and $P_k$ is the $k$th Chebychev polynomial.

In this paper we study the structure of square integrable functionals measurable with respect to coalescing stochastic flows. The case of the Wiener process stopped at the moment of hitting an irregular continuous level is considered. Relying on the change of measure technique, we present a new construction of multiple stochastic integrals with respect to stopped Wiener process. An intrinsic analogue of the Itô–Wiener expansion for the space of square integrable functionals measurable with respect to the stopped Wiener process is constructed.

A family of quasi-invariant measures on the special functional space of curves in a finite-dimensional Euclidean space with respect to the action of diffeomorphisms is constructed. The main result is an explicit expression for the Radon–Nikodym derivative of the transformed measure relative to the original one. The stochastic Ito integral allows to express the result in an invariant form for a wider class of diffeomorphisms. These measures can be used to obtain irreducible unitary representations of the diffeomorphisms group which will be studied in future research. A geometric interpretation of the action considered together with a generalization to the multidimensional case makes such representations applicable to problems of quantum mechanics.

We propose a numerical solution for the solution of the Fokker–Planck–Kolmogorov (FPK) equations associated with stochastic partial differential equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein–Uhlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener–Chaos Expansion. By using this expansion we reformulate the Kolmogorov equation as an infinite system of ordinary differential equations, and by truncating it we set a linear finite system of differential equations. The solution of such system allow us to build an approximation to the solution of the Kolmogorov equations. We test the numerical method with the Kolmogorov equations associated with a stochastic diffusion equation, a Fisher–KPP stochastic equation and a stochastic Burgers equation in dimension 1.

The purpose of this paper is to investigate the harmonic analysis on the time scale ${𝕋}_q$, $q\in (0,1)$ to introduce $q$-weighted Besov spaces subspaces of $L^p({𝕋}_q)$ generalizing the classical one. Further, using an example of $q$-weighted $w_{\alpha ,\beta }(.;q)$ which is introduced and studied. We give a new characterization of the $q$-Besov space using $q$-Poisson kernel and the $g_1$ Littlewood–Paley operator.

Motivated by the direct approach to quantum extensions of Fisher information in Chen and Luo [Front. Math. China 2 (2007) 359], we introduce a further generalized version of Wigner–Yanase–Dyson information, and investigate its fundamental properties including positivity, convexity and super-additivity, etc. As an application, we present a generalized uncertainty relation.