Hu, ZeChun and Ma, ZheMing and Sun, Wei (2006) Extensions of LévyKhintchine Formula and BeurlingDeny Formula in SemiDirichlet Forms Setting. Technical Report. Concordia University. Department of Mathematics & Statistics, Montreal, Quebec.

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Abstract
The LévyKhintchine 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 BeurlingDeny 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 semiDirichlet form. We start with decomposing a regular semiDirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasiregular semiDirichlet forms. Finally, we extend the formulae of LévyKhintchine and BeurlingDeny in semiDirichlet forms setting
through introducing a quasicompatible metric.
Divisions:  Concordia University > Faculty of Arts and Science > Mathematics and Statistics 

Item Type:  Monograph (Technical Report) 
Authors:  Hu, ZeChun and Ma, ZheMing 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:  LévyKhintchine formula; BeurlingDeny formula; Quasiregular semiDirichlet form; Local compactification; Integral representation; Quasicompatible metric 
ID Code:  6671 
Deposited By:  DIANE MICHAUD 
Deposited On:  03 Jun 2010 20:03 
Last Modified:  04 Nov 2016 22:58 
References:
S. Albeverio, Z.M. Ma and M. R¨ockner, A BeurlingDeny type structure theorem for Dirichlet forms on general state space, In: Memorial Volume for R. HøeghKrohn, 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, 1562.
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, Quasihomeomorphism of Dirichlet forms, Nagoya Math. J. 136 (1994) 115.
Ph. Courrège, Sur la forme intégrodifférentielle 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) 673701.
Z. Dong and Z.M. Ma, An integral representation theorem for quasiregular Dirichletspaces, Chinese Sci. Bull. 38 (1993) 13551358.
Z. Dong, Z.M. Ma and W. Sun, A note on BeurlingDeny formulae in infinite dimensional spaces, ACTA Math. Appl. Sinica 13(4) (1997) 353361.
P.J. Fitzsimmons, On the quasiregularity of semiDirichlet forms, Potential Anal. 15(3)(2001) 151182.
M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, BerlinNew York, 1994.
Z.C. Hu and Z.M. Ma, BeurlingDeny formula of semiDirichlet forms, C. R. Math.
Acad. Sci. Paris 338(7) (2004) 521526.
Z.C. Hu, Z.M. Ma and W. Sun, Formulae of BeurlingDeny and LeJan for nonsymmetric Dirichlet forms, preprint, 2005.
Z.C. Hu, BeurlingDeny formula of nonsymmetric Dirichlet forms and the theory of semiDirichlet forms, Ph.D. Dissertation, Sichuan University, 2004.
Z.C. Hu, Some analysis of regular semiDirichlet forms and the associated Hunt processes, preprint, 2004.
[J] N. Jacob, PseudoDifferential Operators and Markov Processes, Vol. 1: Fourier Analysis and Semigroups, Imperial College Press, London, 2001.
J.H. Kim, Stochastic calculus related to nonsymmetric Dirichlet forms, Osaka J. Math. 24 (1987) 331371.
[Ku] K. Kuwae, Functional calculus for Dirichlet forms, Osaka J. Math. 35 (1998) 683715.
S. Mataloni, Representation formulas for nonsymmetric Dirichlet forms, J. Anal. Appl.18(4) (1999) 10391064.
U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994)368421.
Z.M. Ma, L. Overbeck and M. R¨ockner, Markov processes associated with semi Dirichlet forms, Osaka J. Math. 32 (1995) 97119.
Z.M. Ma and M. R¨ockner, Introduction to the Theory of (NonSymmetric) Dirichlet Forms, SpringerVerlag, BerlinHeidelbergNew 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) 7492.
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