Extensions of Lévy-Khintchine formula and Beurling-Deny formula in semi-Dirichlet forms setting |
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Authors: | Ze-Chun Hu Zhi-Ming Ma Wei Sun |
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Affiliation: | a College of Mathematics, Sichuan University, Chengdu 610064, China b Department of Mathematics, Nanjing University, Nanjing 210093, China c Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China d Department of Mathematics and Statistics, Concordia University, Montreal H4B 1R6, Canada |
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Abstract: | The Lévy-Khintchine formula or, more generally, Courrège's theorem characterizes the infinitesimal generator of a Lévy process or a Feller process on Rd. For more general Markov processes, the formula that comes closest to such a characterization is the Beurling-Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Lévy-Khintchine and Beurling-Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric. |
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Keywords: | Lé vy-Khintchine formula Beurling-Deny formula Quasi-regular semi-Dirichlet form Local compactification Integral representation Quasi-compatible metric |
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