Comptes Rendus
no. 7
Potential Theory/Probability Theory
Beurling–Deny formula of semi-Dirichlet forms

Presented by: Paul Malliavin

Ze-Chun Hu1; Zhi-Ming Ma2
1 College of Mathematics, Sichuan University, Chengdu 610064, China
2 Inst. Appl. Math., Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 521-526.

In this Note we announce a structure result for non-symmetric Dirichlet forms and semi-Dirichlet forms. Our result is regarded as an extension of the celebrated Beurling–Deny formula which is up to now available only for symmetric Dirichlet forms. The result can also be regarded as an extension of Lévy–Khinchine formula or more generally, an extension of Courrège's Theorem in the semi-Dirichlet forms setting.

Dans cette Note nous annonçons un résultat de structure pour formes de Dirichlet non-symétrique et demi-forme de Dirichlet. Notre résultat peut être considéré comme une extension de la célèbre formule de Beurling–Deny qui est, jusqu'à maintenant, valable pour formes de Dirichlet symétrique seulement. Le résultat peut être considéré aussi comme une extension de la formule de Lévy–Khinchine, ou plus généralement, une extension du Théorème de Courrège.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.01.020

Ze-Chun Hu 1; Zhi-Ming Ma 2

1 College of Mathematics, Sichuan University, Chengdu 610064, China
2 Inst. Appl. Math., Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China 
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Ze-Chun Hu; Zhi-Ming Ma. Beurling–Deny formula of semi-Dirichlet forms. Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 521-526. doi : 10.1016/j.crma.2004.01.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.01.020/

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[4] J. Bliedtner Dirichlet forms on regular functional spaces, Seminar on Potential Theorey II, Lecture Notes in Math., vol. 226, Springer-Verlag, Berlin, 1971, pp. 15-62

[5] Ph. Courrège Sur la forme intégro-différentielle des opérateurs de CK∞ dans C satisfaisant au principe du maximum, Sém. Théorie du Potentiel, Exposé 2, Volume 38 (1965/1966)

[6] Z. Dong; Z.M. Ma; W. Sun A note on Beurling–Deny formulae in infinite dimensional spaces, Acta Math. Appl. Sinica, Volume 13 (1997) no. 4, pp. 353-361

[7] P.J. Fitzsimmons On the quasi-regularity of semi-Dirichlet forms, Potential Anal., Volume 15 (2001) no. 3, pp. 151-182

[8] M. Fukusima; Y. Oshima; M. Takeda Dirichlet Forms and Symmetric Markov Processes, de Gruyter, 1994

[9] Z.C. Hu, Z.M. Ma, Beurling–Deny formula of non-symmetric Dirichlet forms, Preprint

[10] Z.C. Hu, Z.M. Ma, Extension of Lévy–Khinchine formula in semi-Dirichlet forms setting, in preparation

[11] Z.M. Ma; L. Overbeck; M. Röckner Markov processes associated with semi-Dirichlet forms, Osaka J. Math., Volume 32 (1995), pp. 97-119

[12] Z.M. Ma; M. Röckner Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, Berlin, 1992

[13] K. Sato Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999

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☆ Research supported by 973 project and NSFC.

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