# Vol. 14 (2008), no. 1

### Boundary problems for fully nonlinear parabolic equations with Lévy Laplacian

S. Albeverio, Ya. Belopolskaya, M. Feller

MFAT **14** (2008), no. 1, 1-9

1-9

We suggest a method to solve boundary and initial-boundary value problems for a class of nonlinear parabolic equations with the infinite dimensional L'evy Laplacian $\Delta _L$ $$f\Bigl(U(t,x),\frac{\partial U(t,x)}{\partial t},\Delta_LU(t,x)\Bigl)=0$$ in fundamental domains of a Hilbert space.

### Inverse problem for Stieltjes string damped at one end

Olga Boyko, Vyacheslav Pivovarchik

MFAT **14** (2008), no. 1, 10-19

10-19

Small transversal vibrations of the Stieltjes string, i.e., an elastic thread bearing point masses is considered for the case of one end being fixed and the other end moving with viscous friction in the direction orthogonal to the equilibrium position of the string. The inverse problem of recovering the masses, the lengths of subintervals and the coefficient of damping by the spectrum of vibrations of such a string and its total length is solved.

### On rank one perturbation of continuous spectrum which generates given finite point spectrum

MFAT **14** (2008), no. 1, 20-31

20-31

The perturbations of Nevanlinna type functions which preserve the set of zeros of this function or add to this set new points are discussed.

### Generalized stochastic derivatives on a space of regular generalized functions of Meixner white noise

MFAT **14** (2008), no. 1, 32-53

32-53

We introduce and study generalized stochastic derivatives on a Kondratiev-type space of regular generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to the properties of the stochastic derivatives in the Gaussian analysis. As an example we calculate the generalized stochastic derivative of the solution of some stochastic equation with a Wick-type nonlinearity.

### The involutive automorphisms of $\tau$-compact operators affiliated with a type I von Neuman algebra

K. K. Kudaybergenov, T. S. Kalandarov

MFAT **14** (2008), no. 1, 54-59

54-59

Let $M$ be a type I von Neumann algebra with a center $Z,$ and a faithful normal semi-finite trace $\tau.$ Consider the algebra $L(M, \tau)$ of all $\tau$-measurable operators with respect to $M$ and let $S_0(M, \tau)$ be the subalgebra of $\tau$-compact operators in $L(M, \tau).$ We prove that any $Z$-linear involutive automorphisms of $S_0(M, \tau)$ is inner.

### About nilpotent $C_0$-semigroups of operators in the Hilbert spaces and criteria for similarity to the integration operator

G. V. Lukashenko, G. M. Gubreev

MFAT **14** (2008), no. 1, 60-66

60-66

In the paper, we describe a class of operators $A$ that have empty spectrum and satisfy the nilpotency property of the generated $C_0$-semigroup $U(t)=\exp\{-iAt\},\, t\geqslant 0$, and such that the operator$A^{-1}$ is similar to the integration operator on the corresponding space $L_2(0,a)$.

### A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes

Eugene Lytvynov, Nataliya Ohlerich

MFAT **14** (2008), no. 1, 67-80

67-80

We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space $X$, for which certain fermion point processes are invariant. The Glauber dynamics is a birth-and-death process in $X$, while in the case of the Kawasaki dynamics interacting particles randomly hop over $X$. We establish conditions on generators of both dynamics under which corresponding conservative Markov processes exist.

### Interpolation with a function parameter and refined scale of spaces

Vladimir A. Mikhailets, Aleksandr A. Murach

MFAT **14** (2008), no. 1, 81-100

81-100

The interpolation of couples of separable Hilbert spaces with a function parameter is studied. The main properties of the classical interpolation are proved. Some applications to the interpolation of isotropic Hörmander spaces over a closed manifold are given.