Abstract
We show that a symmetric stable-type form becomes a Dirichlet form in the wide sense under a quite mild assumption and give a necessary and sufficiently condition that the domain contains the family of all uniformly Lipschitz continuous functions with compact support. Moreover we give some path properties of the corresponding Markov processes (we call the processes symmetric stable-like processes) in one dimension such as exceptionality of points and recurrence of the processes. We then note that the recurrence of the processes depend on the behavior of the index functions at the infinity.
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References
Bass, R. F.: 'Occupation time densities for stable-like processes and other pure jump Markov processes', Stochastic Process. Appl. 29 (1988), 65-83.
Bass, R. F.: 'Uniqueness in law for pure jump Markov processes', Probab. Theory Related Fields 79 (1988), 271-287.
Bertoin, J.: Lévy Processes, Cambridge Univ. Press, 1996.
Bretagnolle, J.: 'Résultas de Kesten sur les processus à accroissements indépendents', In: Séminaire de Probabilitée V, Lecture Notes in Math. 191, 1971, pp. 21-36.
Fukushima,M., Oshima, Y. and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, 1994.
Jacob, N.: Pseudo-Differential Operators and Markov Processes, Akademie Verlag, 1996.
Kesten, H.: 'Hitting probabilities of single points for processes with stationary independent increments', Mem. Amer. Math. Soc. 93 (1969).
Komatsu, T.: 'On stable-like processes', In: Proc. of the Seventh Japan-Russia Symp., Held in Tokyo, 1995, Probability Theory and Mathematical Statistics, World Scientific, 1996, pp. 210-219.
Negoro, A.: 'Stable-like processes: Construction of the transition density and the behavior of sample paths near t = 0', Osaka J. Math. 31 (1994), 189-214.
Sato, K.: Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, 1999, originally published in Japanese as Kahou Katei by Kinokuniya, 1990.
Tsuchiya, M.: Lévy measure with generalized polar decomposition and the associated SDE with jumps, Stochastics Stochastics Rep. 38 (1992), 95-117.
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Uemura, T. On Some Path Properties of Symmetric Stable-Like Processes for One Dimension. Potential Analysis 16, 79–91 (2002). https://doi.org/10.1023/A:1024820804141
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DOI: https://doi.org/10.1023/A:1024820804141