Vol. 21 (2015), no. 4

In many models of Mathematical Physics, based on the study of a Markov chain $\widehat \eta= \{\eta_{t}\}_{t=0}^{\infty}$ on ${\mathbb Z}^d$, one can prove by perturbative arguments a contraction property of the stochastic operator restricted to a subspace of local functions $\mathcal H_{M}$ endowed with a suitable norm. We show, on the example of a model of random walk in random environment with mutual interaction, that the condition is enough to prove a Central Limit Theorem for sequences $\{f(S^{k}\widehat \eta)\}_{k=0}^{\infty}$, where $S$ is the time shift and $f$ is strictly local in space and belongs to a class of functionals related to the H\"older continuous functions on the torus $T^{1}$.

With each sequence of real numbers ${\mathbf s}=\{s_j\}_{j=0}^\infty$ two kinds of continued fractions are associated, - the so-called $P-$fraction and a generalized Stieltjes fraction that, in the case when ${\mathbf s}=\{s_j\}_{j=0}^\infty$ is a sequence of moments of a probability measure on $\mathbb R_+$, coincide with the $J-$fraction and the Stieltjes fraction, respectively. A subclass $\mathcal H^{reg}$ of regular sequences is specified for which explicit formulas connecting these two continued fractions are found. For ${\mathbf s}\in\mathcal H^{reg}$ the Darboux transformation of the corresponding generalized Jacobi matrix is calculated in terms of the generalized Stieltjes fraction.

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, and on spaces of regular test and generalized functions of the Levy white noise analysis. In this paper we make the next step: introduce and study operators of stochastic differentiation on spaces of test functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using Lytvynov's generalization of the chaotic representation property. This can be considered as a contribution in a further development of the Levy white noise analysis.

A description of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a reflexive complex Banach space is shown to remain true without the reflexivity requirement. A similar nature description of the entire vectors of exponential type, known for a normal operator in a complex Hilbert space, is generalized to the case of a scalar type spectral operator in a complex Banach space.

The main object of the paper is a Hamiltonian system $J y'-B(t)y=\lambda\Delta(t) y$ defined on an interval $[a,b) $ with the regular endpoint $a$. We define a pseudo\-spectral function of a singular system as a matrix-valued distribution function such that the generalized Fourier transform is a partial isometry with the minimally possible kernel. Moreover, we parameterize all spectral and pseudospectral functions of a given system by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov-Dym and Sakhnovich in this direction.