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

### 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

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 69, no. 10, Oct. 2017, pp. 1406-21, http://umj.imath.kiev.ua/index.php/umj/article/view/1790.

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Section

Research articles