On dual generators for non-local semi-Dirichlet forms

• Autorzy: Uemura T.
• Języki publikacji: EN

Abstrakty

EN

Let k(x, y) be a measurable function defined on E × E off the diagonal, where E is a locally compact separable metric space, and let m be a positive Radon measure on E with full support. In 2012, we showed that a quadratic form having k as a Lévy kernel becomes a lower bounded semi-Dirichlet form on L²(E, m) which is non-local and regular. Then there associates a Hunt process corresponding to the semi-Dirichlet form. In the case where E = R^d, we will show that the dual form of the semi-Dirichlet form also produces a Hunt process by taking a killing. As a byproduct, a precise description of the infinitesimal generator of the dual form is also given.

Słowa kluczowe

EN

semi-Dirichlet form; non-local operator; stable-like process

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2014
• Tom: Vol. 34, Fasc. 2
• Strony: 199--214
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Uemura T.

• Department of Mathematics, Kansai University, Yamate, Suita, Osaka 564-8680, Japan

Bibliografia

• [1] R. F. Bass, Uniqueness in law for pure jump Markov processes, Probab. Theory Related Fields 79 (1988), pp. 271-287.
• [2] S. Carrillo Menendez, Processus de Markov associé à une forme de Dirichlet non symétrique, Z. Wahrsch. Verw. Gebiete 33 (1975), pp. 139-154.
• [3] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, second edition and revised edition, de Gruyter, 2011.
• [4] M. Fukushima and T. Uemura, Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms, Ann. Probab. 40 (2012), pp. 858-889.
• [5] Z.-C. Hu, Z.-M. Ma, and W. Sun, Extensions of Lévy-Khintchine formula and Beurling-Deny formula in semi-Dirichlet forms setting, J. Funct. Anal. 239 (2006), pp. 179-213.
• [6] H. Kunita, Sub-Markov semi-groups in Banach lattices, in: Proceedings of the International Conference on Functional Analysis and Related Topics, University of Tokyo Press, Tokyo 1969, pp. 332-343.
• [7] Z.-M. Ma and M. Röckner, Introduction to the Theory of (Non-symmetric) Dirichlet Forms, Springer, 1992.
• [8] Y. Ōshima, Lectures on Dirichlet spaces, Lecture Notes at Erlangen University, 1988.
• [9] Y. Ōshima, Semi-Dirichlet Forms and Markov Processes, de Gruyter, 2013.
• [10] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.
• [11] R. L. Schilling and T. Uemura, On the Feller property of Dirichlet forms generated by pseudo differential operators, Tohoku Math. J. 59 (2007), pp. 401-422.
• [12] R. L. Schilling and J. Wang, Lower bounded semi-Dirichlet forms associated with Lévy type operators, to appear in: Festschrift Masatoshi Fukushima.
• [13] T. Uemura, On symmetric stable-like processes: some path properties and generators, J. Theoret. Probab. 17 (2004), pp. 541-555.
• [14] T. Uemura, A remark on non-local operators with variable order, Osaka J. Math. 46 (2009), pp. 503-514.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).