Symmetric α-stable stochastic process and the third initial-boundary-value problem for the corresponding pseudodifferential equation
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 a...
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| Date: | 2017 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2017
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1790 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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. |
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