# Construction of a L&eacute;vy-type process by means of the parametrix method

 1Knopova, VP2Kulik, AM1V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv2Institute of Mathematics of the NAS of Ukraine, Kyiv Dopov. Nac. akad. nauk Ukr. 2016, 5:22-29 https://doi.org/10.15407/dopovidi2016.05.022 Section: Information Science and Cybernetics Language: Ukrainian Abstract:  For a wide class of integro-differential operators, it is proved that the $C_\infty({\mathbb R}^n)$-closure of each of such operators is the generator of a semigroup corresponding to a Feller Markov process. The transition probability density of the process is expressed in the form of a convergent series, and the estimates from above and below are provided. The proof is based essentially on a generalization of the parametrix method for the Cauchy problem for pseudodifferential operators. Keywords: generator, Levi's parametrix method, Lévy-type processes, pseudodifferential operator, transition probability density
References:
1. Jacob N. Pseudo-differential operators and Markov processes, I: Fourier analysis and Semigroups, Imperial College Press, 2001. https://doi.org/10.1142/p245
2. Bass R. F. Probab. Th. Rel. Fields., 1988, 79, No 2: 271–287. https://doi.org/10.1007/BF00320922
3. Hoh W. Math. Ann., 1994, 300, No 1: 121–147. https://doi.org/10.1007/BF01450479
4. Hoh, W. Stoch. Stoch. Rep., 1995, 55, No 3–4: 225–252.
5. Komatsu, T. Osaka J. Math., 1984, 21, No 1: 113–132.
6. Tsuchiya M. J. Math. Kyoto Univ., 1970, No 10: 475–492. https://doi.org/10.1215/kjm/1250523730
7. Tsuchiya M. Proc. "Second Japan-USSR Symposium on Probability Theory". Eds. G. Maruyama, Yu. V. Prokhorov Yu., Berlin: Springer, 1972: 490–497.
8. Bass R. F. Probability Surveys., 2004, No 1: 1–19. https://doi.org/10.1214/154957804100000015
9. Jacob N. Pseudo differential operators and Markov processes, III: Markov Processes and Applications, London: Imperial College Press, 2005.
10. Levi E. E. Rend. del. Circ. Mat. Palermo., 1907, 24: 275–317. https://doi.org/10.1007/BF03015067
11. Feller W. Math. Ann., 1936, 113: 113–160. https://doi.org/10.1007/BF01571626
12. Friedman A. Partial differential equations of parabolic type, New-York: Prentice-Hall, 1964.
13. Ethier S. N., Kurtz, T. G. Markov Processes: Characterization and Convergence, New York: Wiley, 1986. https://doi.org/10.1002/9780470316658
14. Jacob N. Pseudo differential operators and Markov processes, II: Generators and their potential theory, London: Imperial College Press, 2002. https://doi.org/10.1142/p264