Portenko N. I.
Antoniouk A. Vict., Berezansky Yu. M., Boichuk A. A., Gutlyanskii V. Ya., Khruslov E. Ya., Kochubei A. N., Korolyuk V. S., Kushnir R. M., Lukovsky I. O., Makarov V. L., Marchenko V. O., Nikitin A. G., Parasyuk I. O., Pastur L. A., Perestyuk N. A., Portenko N. I., Ronto M. I., Sharkovsky O. M., Tkachenko V. I., Trofimchuk S. I.
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 3-6
Symmetric α-stable stochastic process and the third initial-boundary-value problem for the corresponding pseudodifferential equation
Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1406-1421
We consider a pseudodifferential equation of parabolic type with operator of fractional differentiation with respect to a space variable generating a symmetric $\alpha$ -stable process in a multidimensional Euclidean space with an initial condition and a boundary condition imposed on the values of an unknown function at the points of the boundary of a given domain. The last condition is quite similar to the condition of the so-called third (mixed) boundary-value problem in the theory of differential equations with the difference that a traditional (co)normal derivative is replaced in our problem with a pseudodifferential operator. Another specific feature of the analyzed problem is the two-sided character of the boundary condition, i.e., a consequence of the fact that, in the case of \alpha with values between 1 and 2, the corresponding process reaches the boundary making infinitely many visits to both the interior and exterior regions with respect to the boundary.
Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1512-1524
We construct single-layer potentials for a class of pseudodifferential equations connected with symmetric stable stochastic processes. An operator similar to the operator of gradient in the classical potential theory is selected and an analog of the classical theorem on the jump of (co)normal derivative of single-layer potential is established. This result allows us to construct solutions of some initial-boundary-value problems for pseudodifferential equations of the indicated kind.
Berezansky Yu. M., Boichuk A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Perestyuk N. A., Portenko N. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu
Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 3 - 6
Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 859 -864
Berezansky Yu. M., Bojarski B., Gorbachuk M. L., Kopilov A. P., Korolyuk V. S., Lukovsky I. O., Mitropolskiy Yu. A., Portenko N. I., Reshetnyak Yu. G., Samoilenko A. M., Sharko V. V., Shevchuk I. A., Skorokhod A. V., Tamrazov P. M., Zelinskii Yu. B.
Ukr. Mat. Zh. - 2008. - 60, № 5. - pp. 701 – 703
Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1155-1157
Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1158-1162
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.
Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 435-438
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1155-1157
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1272-1282
We consider a multidimensional Wiener process with a semipermeable membrane located on a given hyperplane. The paths of this process are the solutions of a stochastic differential equation, which can be regarded as a generalization of the well-known Skorokhod equation for a diffusion process in a bounded domain with boundary conditions on the boundary. We randomly change the time in this process by using an additive functional of the local-time type. As a result, we obtain a probabilistic representation for solutions of one problem of mathematical physics.
Limit theorems for the number of crossings of a fixed plane by certain sequences of generalized dbffusion processes
Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 357–371
We characterize the weak convergence of certain sequences of generalized diffusion processes by using a specific functional of a process, namely, the number of crossings of a fixed plane by this process.
Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 64 – 74
Ukr. Mat. Zh. - 1974. - 26, № 1. - pp. 25–36
Ukr. Mat. Zh. - 1968. - 20, № 5. - pp. 676–684