Infinite Dimensional Analysis, Quantum Probability and Related Topics

In this paper, we consider least squares type estimators for the drift parameters in the sub-bifractional Vasicek processes defined by $d{X}_t=𝜃(\mu +X_t)dt+d{S}_t^{H,K},t\ge 0,X_0=0$, with unknown parameters $𝜃>0$ and $\mu \in \text{R}$, where $S^{H,K}$ is a sub-bifractional Brownian motion with indices $H\in (0,1)$ and $K\in (0,1]$. The strong consistency results as well as the asymptotic distributions of these estimators are obtained.

In this paper, we investigate the asymptotic behavior of the spatial average of the solution to the parabolic Anderson model with time-independent noise in dimension $d\ge 1$, as the domain of the integral becomes large. We consider three cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index $H\in (\frac{1}{4},\frac{1}{2})$ in dimension d = 1. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.

In this paper, we investigate if the presence of symmetries in the evolution of an open quantum system gives information about the invariant subspaces. Unfortunately, the answer is, in general, negative, but under suitable conditions we can observe that the representation through which we describe the symmetry determines a privileged family of orthogonal projections strongly correlated with the structure of the decoherence-free sub-algebra and, as a consequence, with harmonic projections.

This paper focuses on the average values of functionals like $Y={\int }_0^{1}g(x(t))dt$ on the set $M=\{x|\parallel x{\parallel }_p\le R,x\in C[0,1]\}$ in $L^p[0,1]$. The densities of coordinates of points in M are derived out. The formula of average value EY of functional Y is obtained. The variance DY of Y is proven to be zero, which shows the phenomenon of concentration without measure, and then the nonlinear commutation identity Eh(Y) = h(EY) is obtained for continuous function $h$. Finally, particularly, it is proven that the average value depends on the discretization.

The main purpose of this paper is to derive a general structure of Gegenbauer white noise analysis as a counterpart class of non-Lévy white noise. First, we start with a new detailed construction of the Gegenbauer Fock space ${\mathrm{\Gamma }}_{\beta }({\mathscr{H}})$ which serves to obtain the quantum decomposition associated with the Gegenbauer white noise processes. More precisely, based on the notion of quantum decomposition and the orthogonalization of polynomials of noncommutative Gegenbauer white noise $\omega (t):=b_t^{*}+b_t+b_t^{*}{b}_t{\tilde{b}}_t$, we study the chaos property of the noncommutative $L^2$-space with respect to the vacuum expectation $\tau$. Next, we determine the distribution of the Gegenbauer operator $J({\chi }_D)=〈\omega ,{\chi }_D〉$ and as a consequence we give some useful properties of the Gegenbauer white noise process.

Spaces with asymmetric metric and asymmetric norm are considered. It is shown that any metrizable separable asymmetrically normed linear space $(X,\parallel \cdot |)$ can be isometrically isomorphic imbedded, as an affine linear manifold, into the classical space $C[0,1]$ with uniform norm $\parallel \cdot {\parallel }_C$. A similar result is obtained for spaces of density $𝔞$. For spaces with asymmetric metric, it is shown that each such space of density $𝔞$ is isometric to a part of the space $C({[0,1]}^{𝔞})$ with the asymmetric seminorm $p(f)=\parallel f_{+}{\parallel }_C$, where $f_{+}(t)=max\{f(t),0\}$.