Symmetric α-Stable Stochastic Process and the Third Initial-Boundary-Value Problem for the Corresponding Pseudodifferential Equation

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We consider a pseudodifferential equation of parabolic type with operator of fractional differentiation with respect to a space variable generating a symmetric α-stable process in a multidimensional Euclidean space with initial and boundary conditions imposed on the values of an unknown function at points of the boundary of a given domain. The imposed boundary condition is similar to the condition of the so-called third (mixed) boundary-value problem in the theory of differential equations with the sole difference that the 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 obtained as a consequence of the fact that, in the case of α with values between 1 and 2, the corresponding process reaches the boundary and makes infinitely many visits to both the interior and exterior regions with respect to the boundary.