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Extensions of Lévy-Khintchine Formula and Beurling-Deny Formula in Semi-Dirichlet Forms Setting


Extensions of Lévy-Khintchine Formula and Beurling-Deny Formula in Semi-Dirichlet Forms Setting

Hu, Ze-Chun, Ma, Zhe-Ming and Sun, Wei (2006) Extensions of Lévy-Khintchine Formula and Beurling-Deny Formula in Semi-Dirichlet Forms Setting. Technical Report. Concordia University. Department of Mathematics & Statistics, Montreal, Quebec.

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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.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Monograph (Technical Report)
Authors:Hu, Ze-Chun and Ma, Zhe-Ming and Sun, Wei
Series Name:Department of Mathematics & Statistics. Technical Report No. 1/06
Corporate Authors:Concordia University. Department of Mathematics & Statistics
Institution:Concordia University
Date:February 2006
Keywords:Lévy-Khintchine formula; Beurling-Deny formula; Quasi-regular semi-Dirichlet form; Local compactification; Integral representation; Quasi-compatible metric
ID Code:6671
Deposited On:03 Jun 2010 20:03
Last Modified:18 Jan 2018 17:29


S. Albeverio, Z.M. Ma and M. R¨ockner, A Beurling-Deny type structure theorem for Dirichlet forms on general state space, In: Memorial Volume for R. Høegh-Krohn, Vol.
I, Ideals and Methods in Math. Anal. Stochastics and Appl. (eds. S. Albeverio, J.E. Fenstad, H. Holden and T Lindstrøm), Cambridge University Press, Cambridge, 1992.

J. Bertoin, L´evy Processes, Cambridge University Press, 1996.

J. Bliedtner, Dirichlet forms on regular functional spaces, In: Seminar on Potential Theory II, Lecture Notes in Mathematics No. 226, 1971, 15-62.

Z. Chen, P.J. Fitzsimmons, M. Takeda, J. Ying and T. Zhang, Absolute continuity of symmetric Markov processes, Ann. Probab. 32 (2004) 2067 - 2098.

Z. Chen, Z.M. Ma and M. R¨ockner, Quasi-homeomorphism of Dirichlet forms, Nagoya Math. J. 136 (1994) 1-15.

Ph. Courrège, Sur la forme intégro-différentielle des opérateurs de C1K dans C satisfaisant au principe du maximum, Sém. Théorie du Potentiel, Exposé 2, 38 1965/1966.

Z. Chen and Z. Zhao, Switched diffusion processes and systems of elliptic equations: a Dirichlet space approach, Proc. Royal Edinburgh 124A (1994) 673-701.

Z. Dong and Z.M. Ma, An integral representation theorem for quasi-regular Dirichletspaces, Chinese Sci. Bull. 38 (1993) 1355-1358.

Z. Dong, Z.M. Ma and W. Sun, A note on Beurling-Deny formulae in infinite dimensional spaces, ACTA Math. Appl. Sinica 13(4) (1997) 353-361.

P.J. Fitzsimmons, On the quasi-regularity of semi-Dirichlet forms, Potential Anal. 15(3)(2001) 151-182.

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin-New York, 1994.

Z.C. Hu and Z.M. Ma, Beurling-Deny formula of semi-Dirichlet forms, C. R. Math.
Acad. Sci. Paris 338(7) (2004) 521-526.

Z.C. Hu, Z.M. Ma and W. Sun, Formulae of Beurling-Deny and LeJan for nonsymmetric Dirichlet forms, preprint, 2005.

Z.C. Hu, Beurling-Deny formula of non-symmetric Dirichlet forms and the theory of semi-Dirichlet forms, Ph.D. Dissertation, Sichuan University, 2004.

Z.C. Hu, Some analysis of regular semi-Dirichlet forms and the associated Hunt processes, preprint, 2004.

[J] N. Jacob, Pseudo-Differential Operators and Markov Processes, Vol. 1: Fourier Analysis and Semigroups, Imperial College Press, London, 2001.

J.H. Kim, Stochastic calculus related to non-symmetric Dirichlet forms, Osaka J. Math. 24 (1987) 331-371.
[Ku] K. Kuwae, Functional calculus for Dirichlet forms, Osaka J. Math. 35 (1998) 683-715.

S. Mataloni, Representation formulas for non-symmetric Dirichlet forms, J. Anal. Appl.18(4) (1999) 1039-1064.

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994)368-421.

Z.M. Ma, L. Overbeck and M. R¨ockner, Markov processes associated with semi Dirichlet forms, Osaka J. Math. 32 (1995) 97-119.

Z.M. Ma and M. R¨ockner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, Berlin-Heidelberg-New York, 1992.

K. Sato, L´evy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999.
[Sc] R.L. Schilling, Dirichlet operators and the positive maximum principle, Integr. Equ. Oper. Theory 41 (2001) 74-92.
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