Construction of a L´evy-type process by means of the parametrix method
DOI:
https://doi.org/10.15407/dopovidi2016.05.022Keywords:
transition probability density, L´evy-type processes, pseudodifferential operator, generator, Levi’s parametrix methodAbstract
For a wide class of integro-differential operators, it is proved that the C∞(Rn)-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.
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References
Jacob N. Pseudo-differential operators and Markov processes, I: Fourier analysis and Semigroups, Imperial
College Press, 2001.
Bass R. F. Probab. Th. Rel. Fields., 1988, 79, No 2: 271–287.
Hoh W. Math. Ann., 1994, 300, No 1: 121–147.
Hoh, W. Stoch. Stoch. Rep., 1995, 55, No 3–4: 225–252.
Komatsu, T. Osaka J. Math., 1984, 21, No 1: 113–132.
Tsuchiya M. J. Math. Kyoto Univ., 1970, No 10: 475–492.
Tsuchiya M. Proc. “Second Japan-USSR Symposium on Probability Theory”. Eds. G. Maruyama, Yu. V.
Prokhorov Yu., Berlin: Springer, 1972: 490–497.
Bass R. F. Probability Surveys., 2004, No 1: 1–19.
Jacob N. Pseudo differential operators and Markov processes, III: Markov Processes and Applications,
London: Imperial College Press, 2005.
Levi E. E. Rend. del. Circ. Mat. Palermo., 1907, 24: 275–317.
Feller W. Math. Ann., 1936, 113: 113–160.
Friedman A. Partial differential equations of parabolic type, New-York: Prentice-Hall, 1964.
Ethier S. N., Kurtz, T. G. Markov Processes: Characterization and Convergence, New York: Wiley, 1986.
Jacob N. Pseudo differential operators and Markov processes, II: Generators and their potential theory,
London: Imperial College Press, 2002.
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