Osipchuk M. M.
Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1714-1716
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.
Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1433–1437
The existence of the density of the transition probability is investigated for a generalized diffusion process with transport that satisfies a certain condition of integrability with respect to the Gaussian measure.
Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1224–1230
We construct a generalized diffusion process in a separable Hilbert space. The drift of this process satisfies certain conditions of integrability with respect to the Gaussian measure. We establish the properties of the constructed process.
On a transformation of the wiener process in $ℝ^m$ by a functional of the local time type on a surface
Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 863–866
A transformation of the Wiener process $ξ_t$ in $ℝ^m$ is considered. This transformation is realized by a multiplicative functional $α_l = u(ξ_l/u(ξ_0)$, where the function $u$ is constructed in a certain way by using a functional of the local time type on a surface. It is proved that this transformation is equivalent to the successive application of an absolutely continuous change of a measure and killing on the surface.