Symmetric α-stable stochastic process and the third initial-boundary-value problem for the corresponding pseudodifferential equation

  • M. M. Osipchuk
  • N. I. Portenko

Abstract

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.
Published
25.10.2017
How to Cite
OsipchukM. M., and PortenkoN. I. “Symmetric α-Stable Stochastic Process and the third initial-Boundary-Value Problem for the Corresponding Pseudodifferential Equation”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, no. 10, Oct. 2017, pp. 1406-21, http://umj.imath.kiev.ua/index.php/umj/article/view/1790.
Section
Research articles