# Gasanenko V. A.

### Rarefaction of moving diffusion particles

Gasanenko V. A., Roitman A. B.

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 691-694

We investigate a flow of particles moving along a tube together with gas. The dynamics of particles is determined by a stochastic differential equation with different initial states. The walls of the tube absorb particles. We prove that if the incoming flow of particles is determined by a random Poisson measure, then the number of remained particles is characterized by the Poisson distribution. The parameter of this distribution is constructed by using a solution of the corresponding parabolic boundary-value problem.

### On the Asymptotics of the Sojourn Probability of a Poisson Process between Two Nonlinear Boundaries That Move Away from One Another

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 14-22

We obtain the complete asymptotic decomposition of the sojourn probability of a homogeneous Poisson process inside a domain with curvilinear boundaries. The coefficients of this decomposition are determined by the solutions of parabolic problems with one and two boundaries.

### Complete asymptotic decomposition of the sojourn probability of a diffusion process in thin domains with moving boundaries

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1155–1164

We investigate a diffusion process ξ(*t*) with absorption defined in a thin domain*D* _{ ε }=*{(x,t)∶εG* _{1} *(t) *

_{2}

*(t), t≥0}*. We obtain the complete decomposition of the sojourn probability of ξ(

*t*) in

*D*

_{ε}with respect to ε→0.

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### On certain exact relations for sojourn probabilities of a wiener process

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 842–846

New exact relations are proved for the sojourn probability of a Wiener process between two time-de-pendent boundaries. The proof is based on the investigation of the heat-conduction equation in the domain determined by these functions-boundaries. The relations are given in the form of series.

### A limit theorem for mixing processes subject to rarefaction. II

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 603–612

The limit theorem proved in the first part of this paper is applied to the well-known schemes of processes subject to rarefaction arising in queuing theory, mathematical biology, and in problems for counters.

### A limit theorem for mixing processes subject to rarefaction. I

Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 471–475

We prove a limit theorem on the approximation of point mixing processes subject to rarefaction by general renewal processes. This theorem contains a weaker condition on the mixing coefficient than the known conditions.

### Viability of solutions of many-dimensional stochastic differential equations

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1316–1323

We establish necessary and sufficient conditions for a many-dimensional diffusion process to reside in a fixed domain with probability one.

### Probability for a Wiener process to reside in tube domains for a long period of time

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 912–917

We give various representations of asymptotics for the probability for a Wiener process to reside within a curvilinear strip during extended time intervals.

### Small deviations of solutions of stochastic differential equations in tube domains

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 638–650

We present an algorithm for the determination of a complete asymptotic decomposition of the sojourn probability of a one-dimensional diffusion process in a thin domain with curvilinear boundary.

### A Wiener process in a curvilinear strip

Ukr. Mat. Zh. - 1990. - 42, № 4. - pp. 561–563

### Wiener process in a thin domain

Ukr. Mat. Zh. - 1988. - 40, № 2. - pp. 225-229

### Processes subject to rarefaction

Ukr. Mat. Zh. - 1983. - 35, № 1. - pp. 27—30