非对称狄氏型的Beurling-Deny公式与半狄氏型理论
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摘要
狄氏型(Dirichlet form)理论在经典位势分析与随机分析之间架起了一座桥梁，在位势论，马氏过程，Malliavin分析，量子场论等许多领域有着广泛的应用。对称狄氏型的Beurling-Deny公式在对称狄氏型与对称马氏过程理论里起到了重要作用，比如在考虑对称马氏过程的绝对连续性时，此公式就扮演了重要角色。它使我们更好地了解狄氏型的解析结构，这些解析结构与马氏过程的样本轨道性质有密切联系。虽然有一些数学家试图把公式推广到非对称情形，但迄今Beurling-Deny公式仍然只适用于对称情形。不论是从理论上，还是应用上考察非对称狄氏型的Beurling-Deny公式都有其重要性。本文讨论了非对称狄氏型与半狄氏型的分解问题，给出了一些结构性结果，这些结果可以看作是经典Beurling-Deny公式的推广，也可以看作是Lévy-Khintchine公式或者更广一点是Courrége定理在狄氏型或者半狄氏型框架下的推广；并讨论了一些相关问题。
首先我们讨论了正则非对称狄氏型的Beurling-Deny公式，我们证明可以构造E上与拓扑相容的距离ρ使得对任意a＞0，任意u，v∈C_0(E)∩D(ε)，我们有唯一的分解：
第三章我们为后面的讨论半狄氏型的分解准备一些位势论的工具，主要讨论了正则半狄氏型的有限能量积分测度。
第四章我们讨论了与正则半狄氏型相联系的Hunt过程的一些分析，得到了例外集与ε-例外集的等价性，ε-拟连续函数与q．e．精连续函数的等价性。得到了正则半狄氏型在开集上的部分仍是正则半狄氏型。
第五章我们讨论了半狄氏型的局部紧化与拟同胚，得到了与狄氏型框架下完全类似的结果。
本文最后一章讨论了拟正则半狄氏型的分解问题。在§6.1节我们建立了一个拟正则半狄氏型空间的积分表现定理(定理6．1．1)，此定理在后面讨论拟正则
非对称狄氏型的Beurling一Deny公式与半狄氏型理论
半狄氏型的Be盯lin牙Deny分解时起到关键作用。互6.2节我们讨论了正则半狄氏
型的Beurling一Deny分解.利用虽6.1与吞6.2的结果以及第五章讨论的拟正则半狄
氏型的拟同胚，我们得到了拟正则半狄氏型的分解(定理6.3.1):对于v任D(句
及二任几(v)，有
:(二，。)一/_.2(二(、)一。(x))。(、)，(:二，己、)+关锐(X):(二)、(己x).
J万K石\dJ名
另外，通过引进一个拟相容距离p，我们得到对于一个特殊核D(句。里所有元
素“，v及:>0，我们有(见定理6.3.6)
￡(一，一￡”，‘(一，+关(二，，)。二、二，‘:。(二，，)>。，2(·(、，一(·，，·(、，了(“一“、，
+尤·(X)·(·)、(d·).
关键词:非对称狄氏型，半狄氏型，Beurling一Deny公式，L‘vy一Khintehine公
式，Courr乙ge定理，S.P.V.可积，左强局部性，反vy过程，Hun七过程，扫除
函数，有限能量积分测度，￡一拟连续，q.e.精连续，半狄氏型的部分，局部紧
化，拟同胚，拟支撑，积分表现定理，拟相容距离，核，特殊核，￡一q.e.Lipschitz
函数。
The theory of Dirichlet forms provides a bridge between classical potential theory and stochastic analysis. It has applications in potential theory, Markov processes, Malliavin analysis, quantum field theory, etc. The Beurling-Deny formula of symmetric Dirichlet forms plays an important role in the theory of symmtric Dirichlet forms and symmetric Markov processes. It provides a structure result of the forms which corresponds to the sample path behavior of the corresponding processes. Although, there have been some attempts for the extension of Beurling-Deny formula to non-symmetric case, but up to now it is still available only in symmetric case. This paper discuss the decompositions of non-symmetric Dirichlet forms and semi-Dirichlet forms. We give some structure results which can be regarded as an extension of the classical Beurling-Deny formula, and can also be regarded as an extension of Levy-Khintchine formula or more generally, an extension of Courrege's Theorem in Dirichlet forms or semi-Dirichlet forms setting. We also discuss some related problems.At first, we discuss the Beurling-Deny formula of regular (non-symmetric) Dirichlet forms. We construct a compatible metric p on E such that for any we have the following decomposition:In chapter 3 we prepare some potential tools for the discussions of the decomposition of semi-Dirichlet forms. In this chapter we mainly discuss the measures of finite energy integral with respect to a regular semi-Dirichelt forms.In chapter 4 we study futher some properties of regular semi-Dirichlet forms
and its relation with associated Hunt process. We obtain the equivalence of exceptional sets and ε-exceptional sets, the equivalence of ε-quasi-continuous functions and q.e.-finely continuous functions. We prove that the part of a regular semi-Dirichlet form on an open set is also a regular semi-Dirichlet form, analogous to the case of regular Dirichlet forms.In chapter 5 we discuss the local compactification and quasi-homeomorphism of semi-Dirichlet forms , and get similar results as in Dirichlet forms setting.In final chapter of this paper we study the decompositions of quasi-regular semi-Dirichlet forms. In section 6.1 we establish an integral representation theorem for quasi-regular semi-Dirichlet spaces (cf, Theorem 6.1.1), which plays a key role in the discussion of the Beurling-Deny decompositions of quasi-regular semi-Dirichle forms. In section 6.2 we discuss the Beurling-Deny decompositions of regular semi-Dirichlet forms. By the results in section 6.1, 6.2 and quasi-homeomorphism of quasi-regular semi-Dirichlet forms discussed in chapter 5, we show that for any and we have (cf, Theorem 6.3.1)Moreover, we construct a quasi-compatible metric such that for all u, v in a special core and we have (cf, Theorem 6.3.6)
首先我们讨论了正则非对称狄氏型的Beurling-Deny公式，我们证明可以构造E上与拓扑相容的距离ρ使得对任意a＞0，任意u，v∈C_0(E)∩D(ε)，我们有唯一的分解：
第三章我们为后面的讨论半狄氏型的分解准备一些位势论的工具，主要讨论了正则半狄氏型的有限能量积分测度。
第四章我们讨论了与正则半狄氏型相联系的Hunt过程的一些分析，得到了例外集与ε-例外集的等价性，ε-拟连续函数与q．e．精连续函数的等价性。得到了正则半狄氏型在开集上的部分仍是正则半狄氏型。
第五章我们讨论了半狄氏型的局部紧化与拟同胚，得到了与狄氏型框架下完全类似的结果。
本文最后一章讨论了拟正则半狄氏型的分解问题。在§6.1节我们建立了一个拟正则半狄氏型空间的积分表现定理(定理6．1．1)，此定理在后面讨论拟正则
非对称狄氏型的Beurling一Deny公式与半狄氏型理论
半狄氏型的Be盯lin牙Deny分解时起到关键作用。互6.2节我们讨论了正则半狄氏
型的Beurling一Deny分解.利用虽6.1与吞6.2的结果以及第五章讨论的拟正则半狄
氏型的拟同胚，我们得到了拟正则半狄氏型的分解(定理6.3.1):对于v任D(句
及二任几(v)，有
:(二，。)一/_.2(二(、)一。(x))。(、)，(:二，己、)+关锐(X):(二)、(己x).
J万K石\dJ名
另外，通过引进一个拟相容距离p，我们得到对于一个特殊核D(句。里所有元
素“，v及:>0，我们有(见定理6.3.6)
￡(一，一￡”，‘(一，+关(二，，)。二、二，‘:。(二，，)>。，2(·(、，一(·，，·(、，了(“一“、，
+尤·(X)·(·)、(d·).
关键词:非对称狄氏型，半狄氏型，Beurling一Deny公式，L‘vy一Khintehine公
式，Courr乙ge定理，S.P.V.可积，左强局部性，反vy过程，Hun七过程，扫除
函数，有限能量积分测度，￡一拟连续，q.e.精连续，半狄氏型的部分，局部紧
化，拟同胚，拟支撑，积分表现定理，拟相容距离，核，特殊核，￡一q.e.Lipschitz
函数。
The theory of Dirichlet forms provides a bridge between classical potential theory and stochastic analysis. It has applications in potential theory, Markov processes, Malliavin analysis, quantum field theory, etc. The Beurling-Deny formula of symmetric Dirichlet forms plays an important role in the theory of symmtric Dirichlet forms and symmetric Markov processes. It provides a structure result of the forms which corresponds to the sample path behavior of the corresponding processes. Although, there have been some attempts for the extension of Beurling-Deny formula to non-symmetric case, but up to now it is still available only in symmetric case. This paper discuss the decompositions of non-symmetric Dirichlet forms and semi-Dirichlet forms. We give some structure results which can be regarded as an extension of the classical Beurling-Deny formula, and can also be regarded as an extension of Levy-Khintchine formula or more generally, an extension of Courrege's Theorem in Dirichlet forms or semi-Dirichlet forms setting. We also discuss some related problems.At first, we discuss the Beurling-Deny formula of regular (non-symmetric) Dirichlet forms. We construct a compatible metric p on E such that for any we have the following decomposition:In chapter 3 we prepare some potential tools for the discussions of the decomposition of semi-Dirichlet forms. In this chapter we mainly discuss the measures of finite energy integral with respect to a regular semi-Dirichelt forms.In chapter 4 we study futher some properties of regular semi-Dirichlet forms
and its relation with associated Hunt process. We obtain the equivalence of exceptional sets and ε-exceptional sets, the equivalence of ε-quasi-continuous functions and q.e.-finely continuous functions. We prove that the part of a regular semi-Dirichlet form on an open set is also a regular semi-Dirichlet form, analogous to the case of regular Dirichlet forms.In chapter 5 we discuss the local compactification and quasi-homeomorphism of semi-Dirichlet forms , and get similar results as in Dirichlet forms setting.In final chapter of this paper we study the decompositions of quasi-regular semi-Dirichlet forms. In section 6.1 we establish an integral representation theorem for quasi-regular semi-Dirichlet spaces (cf, Theorem 6.1.1), which plays a key role in the discussion of the Beurling-Deny decompositions of quasi-regular semi-Dirichle forms. In section 6.2 we discuss the Beurling-Deny decompositions of regular semi-Dirichlet forms. By the results in section 6.1, 6.2 and quasi-homeomorphism of quasi-regular semi-Dirichlet forms discussed in chapter 5, we show that for any and we have (cf, Theorem 6.3.1)Moreover, we construct a quasi-compatible metric such that for all u, v in a special core and we have (cf, Theorem 6.3.6)
引文
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[AMR 92] Albeverio S., Ma Z. M. and Rockner M.: A Beurling-Deny type structure theorem for Dirichlet forms on general state space, In: Memorial Volume for R.H0egh- Krohn, Vol.1, Ideals and Methods in Math. Anal. Stochastics and Appl. (eds. Albeverio S., Fenstad J. E., Holden H. and Lindstr0m T. ), Cambridge Univ. Press, Cambridge, 1992.
[AMR 95] Albeverio S., Ma Z. M. and Rockner M.: Characterization of (non-symmetric) semi-Dirichlet forms associated with Hunt processes, Random Oper. and Stoch. Equ., 3(2), 161-179, 1995.
[ARZ 93] Albeverio S., Rockner M. and Zhang T.-S.: Girsanov transform for symmetric diffusions with infinite dimensional state space, Annals of Probability, 2(21), 961- 978, 1993.
[An 72] Ancona A.: Theorie du potentiel dans les espaces fonctionnels a forme coercive, Lecture Notes University Paris VI, 1972.
[An 75] Ancona A.: Sur les espaces de Dirichlet: principes, fonctions de Green, J. Maths pures et appl., 54, 75-124, 1975.
[Ap 02] Applebaum D.: Levy Processes and Stochastic Calculus-Part 1, Preprint, 2002.
[Be 96] Bertoin J.: Levy Processes, Cambridge University Press, 1996.
[BD 59] Beurling A. and Deny J.: Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A., 45, 208-215, 1959.
[B171] Bliedtner J.: Dirichlet forms on regular functional spaces. In: Seminar on Potential Theorey II, Lecture Notes in Math. 226, 15-62, Springer-Verlag, Berlin-Heidelberg- New York, 1971.
[Bl-G 68] Blumenthal R.M. and Getoor R. K.: Markov processes and potential theory, Aca- demic Press, New York and London, 1968.
[BH 91] Bouleau N. and Hirsch F.: Dirichlet forms and analysis on Wiener space. Berlin- New York: de Gruyter, 1991
[Ca-Me 75] Carrillo-Menendez S.: Processus de Markov associe a une forme de Dirichlet non symetrique, Z. Wahrsch. verw. Geb., 33, 139-154, 1975.
[CFTYZ 04] Chen Z.-Q, Fitzsimmons P.J., Takeda M., Ying J.-G. and Zhang T.-S.: Absolute continuity of symmetric Markov processes, to appear in Ann. Probab.
[CMR 94] Chen Z.-Q., Ma Z. M. and Rockner M.: Quasi-homeomorphism of Dirichlet forms, Nagoya Math. J. , 136, 1-15, 1994.
[CZ 02] Chen Z. -Q. and Zhang T. -S.: Girsanov and Feynman-Kac transfomations for symmetric Markov processes, Ann. I. H. Poincare-PR, 4(38), 475-505, 2002.
[CZ 94] Chen Z. -Q. and Zhao Z.: Switched diffusion processes and systems of elliptic equations: a Dirichlet space approach, Proc. Royal Soc. Edinburgh, 124A, 673- 701, 1994.
[CT 78] Chow Y. S. and Teicher H.: Probbility Theory: Independence, Interchaneablity, Martingales, Springer-Verlag, 1978.
[Ch 82] Chung K. L.: Lectures from Markov processes to Brownian motion, Springer- Verlag, 1982.
[Coh 80] Cohn D. L.: Measure Theory, Birkhuser, 1980.
[Cou 66] Courrège PH.: Sur la forme intégro-différentielle des opérateurs de C_K~∞ dans C satisfaisant au principe du maximum, Sém. Théorie du Potentiel, Exposé, 2, 38, 1965/1966.
[D 96] Dong Z.: Some New Results on Dirichlet Forms and Constructions of General Markov Processes, Ph D Dissertation, Beijing, PRC, 1996.
[DM 93] Dong Z. and Ma Z. M.: An integral representation theorem for quasi-regular Dirichlet spaces, Chinese Sci. Bull., 38, 1355-1358, 1993.
[DMS 97] Dong Z., Ma Z. M., Sun W.: A note on Beurling-Deny formulae in infinite dimensional spaces, ACTA Mathematicae Applicatae Sinica, 13(4), 353-361, 1997.
[Fa 92] Fan R. Z.: Beurling-Deny Formulae on Banach Spaces, Acta Mathematica Scientia, 12, 79-84, 1992.
[Fi 97] Fitzsimmons P. J.: Absolute continuity of symmetric diffusions, Annals of Probability, 1(25), 230-258, 1997.
[Fi 01] Fitzsimmons P. J.: On the quasi-regularity of semi-Dirichlet forms, Potential Analysis, 15(3), 151-182, 2001.
[Fu 71] Fukushima M.: Dirichlet spaces and strong Markov processes, Trans. Amer. Math. Soc., 162, 185-224, 1971.
[Fu 80] Fukushima M.: Dirichlet forms and Markov processes, Amsterdam-Oxford-New York: North Holland, 1980.
[Fu 82] Fukushima M.: On absolute continuity of multidimensional symmetrialbe diffusions, Functional Analysis. in Markov processes, Lecture Notes in Math., 923, 146-176, Springer, Berlin, 1982.
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