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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser, Basel, etc. (2004).
N. Jacob, Pseudo-Differential Operators and Markov Processes. Vol. I: Fourier Analysis and Semigroups, Imperial College Press, London (2001); Pseudo-Differential Operators and Markov Processes. Vol. II: Generators and Their Potential Theory, Imperial College Press, London (2002); Pseudo-Differential Operators and Markov Processes. Vol. III: Markov Processes and Applications, Imperial College Press, London (2005).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (1964).
R. M. Blumenthal and R. K. Getoor, “Some theorems on stable processes,” Trans. Amer. Math. Soc., 93, No. 2, 263–273 (1960).
S. Bochner, Lectures on Fourier Integrals [Russian translation], Fizmatgiz, Moscow (1962).
M. M. Osypchuk and M. I. Portenko, “One type of singular perturbations of a multidimensional stable process,” Theory Stochast. Process., 19(35), No. 2, 42–51 (2014).
I. -U. L’obus and M. I. Portenko, “One class of perturbations of a stable process,” Teor. Imovir. Mat. Statyst., No. 52, 102–111 (1995).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 11, pp. 1512–1524, November, 2015.
Rights and permissions
About this article
Cite this article
Osypchuk, M.M., Portenko, M.I. On Simple-Layer Potentials for One Class of Pseudodifferential Equations. Ukr Math J 67, 1704–1720 (2016). https://doi.org/10.1007/s11253-016-1184-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-016-1184-7