# Pogorui A. О.

### The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process

Kolomiets T., Pogorui A. О., Rodriguez-Dagnino R. M.

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 882-889

It is a well-known result that almost all sample paths of a Brownian motion or Wiener process *{W*(*t*)*}* have infinitely many zero-crossings in the interval (0*, δ*) for * δ >* 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of intersections of a level or the number of level-crossings for the telegraph process. Passing to the limit under the Kac condition, we also obtain an estimate of the level-crossings for the Wiener process.

### Multidimensional random motion with uniformly distributed changes of direction and Erlang steps

Pogorui A. О., Rodriguez-Dagnino R. M.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 572-577

In this paper we study transport processes in $\mathbb{R}^n,\quad n \geq 1$, having non-exponential distributed sojourn times or non-Markovian step durations. We use the idea that the probabilistic properties of a random vector are completely determined by those of its projection on a fixed line, and using this idea we avoid many of the difficulties appearing in the analysis of these problems in higher dimensions. As a particular case, we find the probability density function in three dimensions for 2-Erlang distributed sojourn times.

### Fading evolutions in multidimensional spaces

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1577–1582

We study fading random evolutions in multidimensional spaces. By reducing multidimensional cases to the one-dimensional case, we calculate the limit distributions of fading evolutions for some semi-Markov media.

### Asymptotic analysis of phase averaging of a transport process

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 190–198

We investigate asymptotic expansions of solutions of singularly perturbed transport equations in Markov and semi-Markov media.

### Stationary distributions of fading evolutions

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 425-431

We study fading random walks on the line. We determine stationary distributions of the fading Markov evolution and investigate the special semi-Markov case where the sojourn times of the renewal process have Erlang distributions.

### Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 381–387

We determine a stationary measure for a process defined by a differential equation with phase space on the segment $[V_0 , V_1]$ and constant values of a vector field that depend on a controlling semi-Markov process with finite set of states.

### Asymptotic inequalities for the distribution of the time of stay of a semi-Markov process in an expanding set of states

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1586–1590

We establish asymptotic estimates for the behavior of the distribution of the time of the first hit of an infinitely remote level by a semi-Markov process on a semiaxis of integer numbers.

### Limit-ill-posed equations in a Hilbert space

Ukr. Mat. Zh. - 1991. - 43, № 2. - pp. 241–247

The behavior of the solution of a limit-ill-posed problem on fixed compacta is investigated for integral operators, acting in a Hilbert space.