# Volume 57, № 9, 2005

### Volodymyr Semenovych Korolyuk (the 80th anniversary of his birth)

Bratiichuk N. S., Gusak D. V., Kovalenko I. N., Portenko N. I., Samoilenko A. M., Skorokhod A. V.

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1155-1157

### Anatoliy Volodymyrovych Skorokhod (the 75th anniversary of his birth)

Korolyuk V. S., Portenko N. I., Samoilenko A. M., Sytaya G. N.

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1158-1162

### Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits

Pratsiovytyi M. V., Torbin H. M.

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1163–1170

Dedicated to V. S. Korolyuk on occasion of his 80-th birthday

Properties of the set $T_s$ of "particularly nonnormal numbers" of the unit interval are studied in details ($T_s$ consists of real numbers $x$, some of whose $s$-adic digits have the asymptotic frequencies in the nonterminating $s$-adic expansion of $x$, and some do not).
It is proven that the set $T_s$ is residual in the topological sense (i.e., it is of the first Baire category)
and it is generic in the sense of fractal geometry ( $T_s$ is a superfractal set, i.e., its Hausdorff - Besicovitch dimension is equal to 1).
A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their $s$-adic expansions is presented.

### Topological Spaces with Skorokhod Representation Property

Banakh T. O., Bogachev V. I., Kolesnikov A. V.

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1171–1186

We give a survey of recent results that generalize and develop a classical theorem of Skorokhod on representation of weakly convergent sequences of probability measures by almost everywhere convergent sequences of mappings.

### Weakly Sub-Gaussian Random Elements in Banach Spaces

Kvaratskheliya V. V., Tarieladze V. I., Vakhaniya N. N.

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1187–1208

We give a survey of properties of weakly sub-Gaussian random elements in infinite-dimensional spaces. Some new results and examples are also given.

### On the Exit of One Class of Random Walks from an Interval

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1209–1217

We consider the random walk $S_n = \sum_{k\leqn}\xi_k \quad (S_n = 0)$ whose characteristic function of jumps $\xi_k$ satisfies the condition of almost semicontinuity. We investigate the problem of the exit of such $S_n$ from a finite interval.

### Smoothing Problem in Anticipating Scenario

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1218–1234

We consider a smoothing problem for stochastic processes satisfying stochastic differential equations with Wiener processes that may not have a semimartingale property with respect to the joint filtration.

### Stochastic Systems with Averaging in the Scheme of Diffusion Approximation

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1235–1252

We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by jump Markov and semi-Markov processes. We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the diffusion approximation using the asymptotic decomposition of generating operators and solutions of problems of singular perturbation for reducibly inverse operators.

### Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1253–1260

We consider an evolutionary system switched by a semi-Markov process. For this system, we obtain an inhomogeneous diffusion approximation results where the initial process is compensated by the averaging function in the average approximation scheme.

### On the Regularity of Distribution for a Solution of SDE of a Jump Type with Arbitrary Levy Measure of the Noise

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1261–1283

The local properties of distributions of solutions of SDE's with jumps are studied. Using the method based on the “time-wise” differentiation on the space of functionals of Poisson point measure, we give a full analog of Hormander condition, sufficient for the solution to have a regular distribution. This condition is formulated only in terms of coefficients of the equation and does not require any regularity properties of the Levy measure of the noise.

### Stochastic and Deterministic Bundles

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1284–1288

We consider a bundle determined by a classifying map with skeleton smooth in the Chen — Souriau sense. We show that the stochastic classifying map is homotopic to a deterministic classifying map on the Holder loop space.

### Measure-Valued Diffusions and Continual Systems of Interacting Particles in a Random Medium

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1289–1301

We consider continual systems of stochastic equations describing the motion of a family of interacting particles whose mass can vary in time in a random medium. It is assumed that the motion of every particle depends not only on its location at given time but also on the distribution of the total mass of particles. We prove a theorem on unique existence, continuous dependence on the distribution of the initial mass, and the Markov property. Moreover, under certain technical conditions, one can obtain the measure-valued diffusions introduced by Skorokhod as the distributions of the mass of such systems of particles.

### On Renewal Equations Appearing in Some Problems in the Theory of Generalized Diffusion Processes

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1302–1312

We construct a Wiener process on a plane with semipermeable membrane located on a fixed circle and acting in the normal direction. The construction method takes into account the symmetry properties of both the circle and the Wiener process. For this reason, the method is reduced to the perturbation of a Bessel process by a drift coefficient that has the type of a δ-function concentrated at a point. This leads to a pair of renewal equations, using which we determine the transition probability of the radial part of the required process.