# Volume 49, № 3, 1997

### Yurii L’vovich Daletskii

Berezansky Yu. M., Korolyuk V. S., Krein S. G., Mitropolskiy Yu. A., Samoilenko A. M., Skorokhod A. V.

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 323–325

### Infinite systems of stochastic differential equations and some lattice models on compact Riemannian manifolds

Daletskii A. Yu., Kondratiev Yu. G.

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 326–337

Stochastic dynamics associated with Gibbs measures on an infinite product of compact Riemannian manifolds is constructed. The probabilistic representations for the corresponding Feller semigroups are obtained. The uniqueness of the dynamics is proved.

### A priori estimates for solutions of the first initial boundary-value problem for systems of fully nonlinear partial differential equations

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 338–363

We prove a priori estimates for a solution of the first initial boundary-value problem for a system of fully nonlinear partial differential equations (PDE) in a bounded domain. In the proof, we reduce the initial boundary-value problem to a problem on a manifold without boundary and then reduce the resulting system on the manifold to a scalar equation on the total space of the corresponding bundle over the manifold.

### Infinite-dimensional analysis related to generalized translation operators

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 364–409

We give an extensive generalization of the white-noise analysis (in the Gaussian and non-Gaussian case) in which the role of translation operators is played by a fixed family of generalized translation operators.

### Admissible vector fields and related diffusions on infinite-dimensional manifolds

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 410–423

A variation on the notion of “admissibility” for vector fields on certain infinite-dimensional manifolds with measures on them is described. It leads to the construction of associated diffusions and Markov semigroups on these manifolds via Dirichlet forms. Some classes of concrete examples are given.

### One class of solutions of Volterra equations with regular singularity

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 424–432

The Volterra integral equation of the second order with a regular singularity is considered. Under the conditions that a kernel *K(x,t)* is a real matrix function of order *n×n* with continuous partial derivatives up to order *N*+1 inclusively and *K*(0,0) has complex eigenvalues ν±*i* μ (ν>0), it is shown that if ν>2|‖*K*|‖_{ C }-*N*-1, then a given equation has two linearly independent solutions.

### Boundary-value problems for stationary Hamilton-Jacobi and Bellman equations

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 433–447

We introduce solutions of boundary-value problems for the stationary Hamilton-Jacobi and Bellman equations in functional spaces (semimodules) with a special algebraic structure adapted to these problems. In these spaces, we obtain representations of solutions in terms of “basic” ones and prove a theorem on approximation of these solutions in the case where nonsmooth Hamiltonians are approximated by smooth Hamiltonians. This approach is an alternative to the maximum principle.

### Regularity results for Kolmogorov equations in $L^2 (H, μ)$ spaces and applications

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 448–457

We consider the transition semigroup $R_t =e^{tsA}$ associated to an Ornstein—Uhlenbeck process in a Hilbert space $H$. We characterize, under suitable assumptions, the domain of $A$ as a subspace $W^{2,2} (H, μ)$, where $μ$ is the invariant measure associated to $R_t$. This characterization is then used to treat some Kolmogorov equations with variable coefficients.

### Measure-valued diffusion

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 458–464

We consider the class of continuous measure-valued processes {*μ* _{ t }} on a finite-dimensional Euclidean space *X* for which ∫*fd* *μ* _{ t } is a semimartingale with absolutely continuous characteristics with respect to *t* for all *f:X→R* smooth enough. It is shown that, under some general condition, the Markov process with this property can be obtained as a weak limit for systems of randomly interacting particles that are moving in *X* along the trajectories of a diffusion process in *X* as the number of particles increases to infinity.

### On Hausdorff-Young inequalities for quantum fourier transformations

Ukr. Mat. Zh. - 1997. - 49, № 3. - pp. 465–471

The classical Hausdorff-Young inequality for the Fourier transformation is generalized to various quantum contexts involving noncommutative *L* ^{ p }-spaces based on translation-invariant traces.