保正半群的hh-变换，广义狄氏型的扰动及相关问题的研究
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摘要
从1957年Doob, L. J考虑并构造条件布朗运动开始(见[30]),Doob-h-变换一直是很多学者关心的问题(见[6,25,32,36,62,65,66,81]等及其参考文献)。任给一对L2(E；m)上的有保正性的强连续压缩对偶半群(Tt)t>0和(Tt)t>0,通过它们的一对α-过分函数h及共轭-α-过分函数h对它们做hh-变换,本文得到一对L2(E;hh·m)上的强连续压缩的次马氏半群(Tt/h)t>0和(Tt/h)t>0(见定理2.2.1,定理2.2.2),并且得到它们在新的空间L2(E；hh.m)上是对偶的(见定理2.2.3),还分析了它们的无穷小生成元(见命题2.2.1及命题2.2.2)。特别地,如果(Tt)t>o和(Tt)t>0中的一个(不妨设(Tt)t>0)有次马氏性,则我们得到一对L2(E；h.m)上的对偶的强连续压缩次马氏半群(e-atTt)t>0和(Tt/h)t>0(见定理2.2.4)。最后在拟正则狄氏型框架下,本文给出了(Tt)t>0和(Tt)t>0在一定意义下结合一对右过程的充要条件为其联系的保正型(￡,D(ε))是拟正则的,进而得到这对右过程在新的测度hh.m下是弱对偶的,并且满足一般的Hunt-假设：从几乎所有点出发都不能立即到达的集合永远不能到达(即是ε-例外集)(见定理2.2.6)。特别地,当(Tt)t>0和(Tt)t>0联系着一个拟正则半狄氏型时,也有类似的结果(见定理2.2.7)。
拟正则狄氏型与右过程的一一对应关系(见[14,52,53]等),为研究经典的位势论及随机分析提供了一个有力的工具。狄氏型的扰动以及与之紧密联系的算子扰动以及广义Feynman-Kac半群等,一直是国际上狄氏型及其相关领域的一个研究热点。而有关广义狄氏型的符号光滑测度扰动却一直没多少人讨论,在这个问题上本文做了一些探讨,给出了扰动后的二次型仍然是广义狄氏型的几个充分条件(见定理3.2.1,定理3.2.2,定理3.2.3),以及扰动后的广义狄氏型结合马氏过程的充分条件(见定理3.2.5)。特别地,本文还研究了非对称狄氏型扰动的相关问题。任给一个非对称的拟正则狄氏型,本文研究了一类特殊的位势项,得到它们是拟连续的,并且刻画了这类特殊的位势项在扰动后狄氏型定义域中的充分条件(见定理3.3.1)；然后利用这一结果直接证明了两个常用的转换等式(见命题3.3.1)。
在最后一章中,本文综合利用h-变换,狄氏型扰动以及Girsanov变换三种方法,刻画了布朗运动零能量可加泛函的渐近性(见定理4.2.3)。
下面我们按章节顺序简单叙述一下本文的主要内容。
第一章第一节叙述本文的研究背景及主要研究结果；第二节介绍一些基本概念及一些已有的结论。
第二章第一节给出保正型的h-变换的一些最新结果并引入本文要解决的问题。第二节首先定义了L2(E；m)空间上保正半群(Tt)t>0和(Tt)t>0的hh-变换：容易验证变换后的半群都有次马氏性,然后我们证明了(Tth)t>0和(Tth)t>0分别是L1(E;hh-m)和L∞(E;hh·m)上的压缩算子(见命题2.2.1),再利用Riesz-Thorin插值定理,得到它们都是L2(E;hh·m)上的压缩算子(见命题2.2.1),进而证明它们都是L2(E;hh·m)上的强连续压缩的对偶半群(见定理2.2.1,定理2.2.2,定理2.2.3)。最后在拟正则狄氏型框架下,本文给出了(Tt)t>0和(Tt)t>0在一定意义下结合一对对偶右过程的充要条件(见定理2.2.6)。特别地,对于一对与拟正则半狄氏型结合的半群,我们得到了类似的结果。
第三章研究广义狄氏型的扰动,并探讨了非对称狄氏型扰动后的一类位势项的性质。第一节,我们简单叙述广义狄氏型以及非对称狄氏型扰动的研究背景。第二节主要研究如下形式的广义狄氏型的扰动：得到当μ属于Hardy-类的光滑测度时,扰动后的二次型εμ的定义域以及范数保持不变(见引理3.2.1),且εμ仍然是广义狄氏型(见定理3.2.1)；还给出了当μ是符号光滑测度时,εμ是广义狄氏型的充分条件(见定理3.2.3),及这个广义狄氏型结合右过程的一个充分条件(见定理3.2.5)。在第三节中,设(X,X)为与拟正则的(非对称)狄氏型(ε,D(ε))联系的一对对偶的马氏过程。μ为光滑测度,对任意的f∈L2(E；μ),任意实数α,p>0,定义如下位势项：我们得到UAα+pμf以及UAα+pμf是拟连续的并且在扰动后的狄氏型定义域D(εμ)中(见定理3.3.1)；进而利用狄氏型理论得到了两个转换等式(见命题3.3.1),并给出了该结果的一个应用(见命题3.3.2)：当一对对偶过程(X,X)联系着一个狄氏型时,任给一个光滑测度μ及其唯一对应的正的连续可加泛函At以及At,(Y,Y)是分别由At以及At诱导的(X,X)的时间变换过程,则Y和Y是L2(E；μ)上的一对对偶的右过程。
第四章主要给出h-变换以及狄氏型的扰动在广义Feynman-Kac泛函的渐近性以及对偶过程时间变换等方面的应用,得到如下结果(见定理4.2.3)：如果Lt/-u“是鞅,u有界,那么对任意的x∈Rd有这里
The well-known Doob- h-transform has been researched by many researchers (cf. [6,25,32,36,62,65,66,81] and the references therein) since the construction of conditional Brownian motion by Doob, L. J. in 1957 (cf. [30]). In this dissertation, given a-excessive h and co-a-excessive h we consider the hh-transforms of strongly continuous contraction dual semigroups (Tt)t>o and (Tt)t>o on L2(E;m), which are only positivity preserving semigroups, and get strongly continuous contraction semi-groups (Tth)t>0 and (Tth)t>0 with sub-Markovian property on L2(E:hh·m). More-over, we show that (Tth)t>0 and (Tth)t>0 are in duality in the space L2(E;hh·m). In particular, if one (say (Tt)t>0) of the dual semigroups (Tt)t>o and (Tt)t>0 is sub-Markovian, we can also get a pair of sub-Markovian dual semigroups (e-αtTt)t>0 and (Tth)t>0 on L2(E;h·m) (cf. corollary 2.2.4). Finally, under the framework of quasi-regular Dirichlet forms, we give a necessary and sufficient condition for (Tt)t>0 and (Tt)t>0 to be hh-associated with a pair of special standard processes which are in duality. Moreover, those right processes satisfy Hunt's Hypothesis:if a set can not be reached immediately from almost every point, then it can not be reached forever, e.g. be aε-exceptional set). In particular, we obtain the similar result:for a pair of semigroups associated with a quasi-regular semi-Dirichlet form (cf. Theorem 2.2.7).
The one-to-one correspondence between quasi-regular Dirichlet forms and right processes (cf. [14,52,53]) is a powerful tool in studying the classical potential theory and stochastic analysis. Although perturbation of Dirichlet forms has been studied intensively by many researchers, there are few researchers who consider the perturbation of generalized Dirichlet forms by signed smooth measures. For this kind of perturbation, we get several sufficient conditions for the perturbed form to be a generalized Dirichlet form (cf. Theorem 3.2.1, Theorem 3.2.2 and Theorem 3.2.3), and give one sufficient condition for the perturbed generalized Dirichlet form to be associated with a Markov process (cf. Theorem 3.2.5).
Next we will focus on the potential terms of the perturbed Dirichlet forms, give one sufficient condition for them to be in the domain of the perturbed Dirichlet forms and are quasi-continuous. Finally, we prove two switching identities directly by perturbation of non-symmetric Dirichlet forms and the above results.
In the last Chapter, by h-transform, perturbation of Dirichlet forms and Gir-sanov transform, we get some results about the asymptotic property of the additive functional of zero energy for Brownian motion (cf. Theorem 4.2.3).
In details the contents of this dissertation are organized as follows.
In the first section of Chapter I, we recall some background and give the main results of this dissertation. In the second section of this Chapter, we introduce some necessary concepts. In Chapter II, we start with h-transforms of positivity preserving forms and state some recent results about h-transforms in the first section. In the second section, we first define the hh-transforms of positivity preserving dual semigroups (Tt)t>0 and (Tt)t>0 on L2(E;m) (cf. Proposition 2.2.1), and show that (Tth)t>o and (Tth)t>0 are contraction operators on L1(E;hh·m) and L∞(E;hh m), respectively (cf. Proposition 2.2.1). Then by Riesz-Thorin theory we get that they are contraction operators on L2 (E:hh·m) and prove that they are strongly continuous contraction semigroups on L2(E;hh·m) (cf. Theorem 2.2.1, Theorem 2.2.2). In particular, if (Tt)t>0 is sub-Markovian, we can also get a pair of sub-Markovian dual semigroups (e-atTt)t>0 and (Tth)t>0 on L2(E;h·m) (cf. corollary 2.2.4). Finally, under the framework of quasi-regular Dirichlet form, we give a necessary and sufficient condition for (Tt)t>0 and (Tt)t>o to be associated with a pair of right processes in some sense (cf. Theorem 2.2.6). In particular, if the pair of semigroups is associated with a quasi-regular semi-Dirichlet form, we can show that they are also associated with a pair of dual Markov processes on L2(E;h·m) (cf. Theorem 2.2.7).
In Chapter III, we mainly consider the perturbation of generalized Dirichlet forms and focus on the properties of certain potential items of the perturbed non-symmetric Dirichlet forms. In the first section, we consider the generalized Dirichlet form as follows: we show that ifμis a smooth measure in Hardy-class, then the domain and the norm of the perturbed formεμare the same as that of the formε(cf. Theorem 3.2.1) and are still generalized Dirichlet forms. Also, we give some conditions for the perturbed formεμto be generalized Dirichlet forms (cf. Theorem 3.2.1, Theorem 3.2.2), whenμis a signed smooth measure. Moreover, we give one sufficient condition for the perturbed generalized Dirichlet formεμto be associated with a right process. In the second section, given a quasi-regular Dirichlet form (ε,D(ε)) and the associated dual processes (X,X), a smooth measureμ, we consider the following potential terms, We get that if f∈L2(E;μ), a>0,p>0, then UAα+pμf, UAα+pμf∈D(εμ) and are quasi-continuous (cf. Theorem 3.3.1). Then by the theory of Dirichlet forms we give two switching identities directly (cf. Theorem 3.3.1), and finally we give an application of the switching identities (cf. Proposition 3.3.2):for a pair of dual processes (X, X) which are associated with a quasi-regular Dirichlet form, let At At be the positive continuous additive functional whose revuz measure isμ, (Y, Y) be the time-change processes of (X,X) by At and At, then Y and Y are a pair of dual right processes on L2(E;μ).
In the last Chapter, we give some applications of h-transform and perturbation of Dirichlet forms in. Feynman-Kac functionals and the time-change processes of dual processes. The main results are as follows (cf. Theorem 4.2.3):If Lt-u is a martingale, u is bounded,▽u∈Kd-1 and ||E.[eMt-u]||9<∞, then for any x∈Rd, we have Here D(ε)b=D(ε) n L∞(Rd, dx).
拟正则狄氏型与右过程的一一对应关系(见[14,52,53]等),为研究经典的位势论及随机分析提供了一个有力的工具。狄氏型的扰动以及与之紧密联系的算子扰动以及广义Feynman-Kac半群等,一直是国际上狄氏型及其相关领域的一个研究热点。而有关广义狄氏型的符号光滑测度扰动却一直没多少人讨论,在这个问题上本文做了一些探讨,给出了扰动后的二次型仍然是广义狄氏型的几个充分条件(见定理3.2.1,定理3.2.2,定理3.2.3),以及扰动后的广义狄氏型结合马氏过程的充分条件(见定理3.2.5)。特别地,本文还研究了非对称狄氏型扰动的相关问题。任给一个非对称的拟正则狄氏型,本文研究了一类特殊的位势项,得到它们是拟连续的,并且刻画了这类特殊的位势项在扰动后狄氏型定义域中的充分条件(见定理3.3.1)；然后利用这一结果直接证明了两个常用的转换等式(见命题3.3.1)。
在最后一章中,本文综合利用h-变换,狄氏型扰动以及Girsanov变换三种方法,刻画了布朗运动零能量可加泛函的渐近性(见定理4.2.3)。
下面我们按章节顺序简单叙述一下本文的主要内容。
第一章第一节叙述本文的研究背景及主要研究结果；第二节介绍一些基本概念及一些已有的结论。
第二章第一节给出保正型的h-变换的一些最新结果并引入本文要解决的问题。第二节首先定义了L2(E；m)空间上保正半群(Tt)t>0和(Tt)t>0的hh-变换：容易验证变换后的半群都有次马氏性,然后我们证明了(Tth)t>0和(Tth)t>0分别是L1(E;hh-m)和L∞(E;hh·m)上的压缩算子(见命题2.2.1),再利用Riesz-Thorin插值定理,得到它们都是L2(E;hh·m)上的压缩算子(见命题2.2.1),进而证明它们都是L2(E;hh·m)上的强连续压缩的对偶半群(见定理2.2.1,定理2.2.2,定理2.2.3)。最后在拟正则狄氏型框架下,本文给出了(Tt)t>0和(Tt)t>0在一定意义下结合一对对偶右过程的充要条件(见定理2.2.6)。特别地,对于一对与拟正则半狄氏型结合的半群,我们得到了类似的结果。
第三章研究广义狄氏型的扰动,并探讨了非对称狄氏型扰动后的一类位势项的性质。第一节,我们简单叙述广义狄氏型以及非对称狄氏型扰动的研究背景。第二节主要研究如下形式的广义狄氏型的扰动：得到当μ属于Hardy-类的光滑测度时,扰动后的二次型εμ的定义域以及范数保持不变(见引理3.2.1),且εμ仍然是广义狄氏型(见定理3.2.1)；还给出了当μ是符号光滑测度时,εμ是广义狄氏型的充分条件(见定理3.2.3),及这个广义狄氏型结合右过程的一个充分条件(见定理3.2.5)。在第三节中,设(X,X)为与拟正则的(非对称)狄氏型(ε,D(ε))联系的一对对偶的马氏过程。μ为光滑测度,对任意的f∈L2(E；μ),任意实数α,p>0,定义如下位势项：我们得到UAα+pμf以及UAα+pμf是拟连续的并且在扰动后的狄氏型定义域D(εμ)中(见定理3.3.1)；进而利用狄氏型理论得到了两个转换等式(见命题3.3.1),并给出了该结果的一个应用(见命题3.3.2)：当一对对偶过程(X,X)联系着一个狄氏型时,任给一个光滑测度μ及其唯一对应的正的连续可加泛函At以及At,(Y,Y)是分别由At以及At诱导的(X,X)的时间变换过程,则Y和Y是L2(E；μ)上的一对对偶的右过程。
第四章主要给出h-变换以及狄氏型的扰动在广义Feynman-Kac泛函的渐近性以及对偶过程时间变换等方面的应用,得到如下结果(见定理4.2.3)：如果Lt/-u“是鞅,u有界,那么对任意的x∈Rd有这里
The well-known Doob- h-transform has been researched by many researchers (cf. [6,25,32,36,62,65,66,81] and the references therein) since the construction of conditional Brownian motion by Doob, L. J. in 1957 (cf. [30]). In this dissertation, given a-excessive h and co-a-excessive h we consider the hh-transforms of strongly continuous contraction dual semigroups (Tt)t>o and (Tt)t>o on L2(E;m), which are only positivity preserving semigroups, and get strongly continuous contraction semi-groups (Tth)t>0 and (Tth)t>0 with sub-Markovian property on L2(E:hh·m). More-over, we show that (Tth)t>0 and (Tth)t>0 are in duality in the space L2(E;hh·m). In particular, if one (say (Tt)t>0) of the dual semigroups (Tt)t>o and (Tt)t>0 is sub-Markovian, we can also get a pair of sub-Markovian dual semigroups (e-αtTt)t>0 and (Tth)t>0 on L2(E;h·m) (cf. corollary 2.2.4). Finally, under the framework of quasi-regular Dirichlet forms, we give a necessary and sufficient condition for (Tt)t>0 and (Tt)t>0 to be hh-associated with a pair of special standard processes which are in duality. Moreover, those right processes satisfy Hunt's Hypothesis:if a set can not be reached immediately from almost every point, then it can not be reached forever, e.g. be aε-exceptional set). In particular, we obtain the similar result:for a pair of semigroups associated with a quasi-regular semi-Dirichlet form (cf. Theorem 2.2.7).
The one-to-one correspondence between quasi-regular Dirichlet forms and right processes (cf. [14,52,53]) is a powerful tool in studying the classical potential theory and stochastic analysis. Although perturbation of Dirichlet forms has been studied intensively by many researchers, there are few researchers who consider the perturbation of generalized Dirichlet forms by signed smooth measures. For this kind of perturbation, we get several sufficient conditions for the perturbed form to be a generalized Dirichlet form (cf. Theorem 3.2.1, Theorem 3.2.2 and Theorem 3.2.3), and give one sufficient condition for the perturbed generalized Dirichlet form to be associated with a Markov process (cf. Theorem 3.2.5).
Next we will focus on the potential terms of the perturbed Dirichlet forms, give one sufficient condition for them to be in the domain of the perturbed Dirichlet forms and are quasi-continuous. Finally, we prove two switching identities directly by perturbation of non-symmetric Dirichlet forms and the above results.
In the last Chapter, by h-transform, perturbation of Dirichlet forms and Gir-sanov transform, we get some results about the asymptotic property of the additive functional of zero energy for Brownian motion (cf. Theorem 4.2.3).
In details the contents of this dissertation are organized as follows.
In the first section of Chapter I, we recall some background and give the main results of this dissertation. In the second section of this Chapter, we introduce some necessary concepts. In Chapter II, we start with h-transforms of positivity preserving forms and state some recent results about h-transforms in the first section. In the second section, we first define the hh-transforms of positivity preserving dual semigroups (Tt)t>0 and (Tt)t>0 on L2(E;m) (cf. Proposition 2.2.1), and show that (Tth)t>o and (Tth)t>0 are contraction operators on L1(E;hh·m) and L∞(E;hh m), respectively (cf. Proposition 2.2.1). Then by Riesz-Thorin theory we get that they are contraction operators on L2 (E:hh·m) and prove that they are strongly continuous contraction semigroups on L2(E;hh·m) (cf. Theorem 2.2.1, Theorem 2.2.2). In particular, if (Tt)t>0 is sub-Markovian, we can also get a pair of sub-Markovian dual semigroups (e-atTt)t>0 and (Tth)t>0 on L2(E;h·m) (cf. corollary 2.2.4). Finally, under the framework of quasi-regular Dirichlet form, we give a necessary and sufficient condition for (Tt)t>0 and (Tt)t>o to be associated with a pair of right processes in some sense (cf. Theorem 2.2.6). In particular, if the pair of semigroups is associated with a quasi-regular semi-Dirichlet form, we can show that they are also associated with a pair of dual Markov processes on L2(E;h·m) (cf. Theorem 2.2.7).
In Chapter III, we mainly consider the perturbation of generalized Dirichlet forms and focus on the properties of certain potential items of the perturbed non-symmetric Dirichlet forms. In the first section, we consider the generalized Dirichlet form as follows: we show that ifμis a smooth measure in Hardy-class, then the domain and the norm of the perturbed formεμare the same as that of the formε(cf. Theorem 3.2.1) and are still generalized Dirichlet forms. Also, we give some conditions for the perturbed formεμto be generalized Dirichlet forms (cf. Theorem 3.2.1, Theorem 3.2.2), whenμis a signed smooth measure. Moreover, we give one sufficient condition for the perturbed generalized Dirichlet formεμto be associated with a right process. In the second section, given a quasi-regular Dirichlet form (ε,D(ε)) and the associated dual processes (X,X), a smooth measureμ, we consider the following potential terms, We get that if f∈L2(E;μ), a>0,p>0, then UAα+pμf, UAα+pμf∈D(εμ) and are quasi-continuous (cf. Theorem 3.3.1). Then by the theory of Dirichlet forms we give two switching identities directly (cf. Theorem 3.3.1), and finally we give an application of the switching identities (cf. Proposition 3.3.2):for a pair of dual processes (X, X) which are associated with a quasi-regular Dirichlet form, let At At be the positive continuous additive functional whose revuz measure isμ, (Y, Y) be the time-change processes of (X,X) by At and At, then Y and Y are a pair of dual right processes on L2(E;μ).
In the last Chapter, we give some applications of h-transform and perturbation of Dirichlet forms in. Feynman-Kac functionals and the time-change processes of dual processes. The main results are as follows (cf. Theorem 4.2.3):If Lt-u is a martingale, u is bounded,▽u∈Kd-1 and ||E.[eMt-u]||9<∞, then for any x∈Rd, we have Here D(ε)b=D(ε) n L∞(Rd, dx).
引文
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[9]陈传钟,孙玮.狄氏型的扰动及其结合的马氏过程[J].数学物理学报21A(2)(2001)：154-161
[10]Chen, C. Z. and Sun, W., Strong continuity of Generalized Feynman-Kac Semigroups: Necessary and Sufficient conditions [J]. J. Funct. Anal.237(2006):446-465
[11]陈传钟.广义狄氏型的扰动及其结合的马氏过程[J].应用数学学报24(4)(2001)：561-567
[12]Chen, C. Z.,狄氏型的扰动与Feynman-Kac半群[D].中南大学博士学位论文2004
[13]Chen,C. Z., A Note on Perturbation of Non-Symmetric Dirichlet Forms by Signed Smooth Measures [J]. Acta Mathematica Scientia.27B(1)(2007):119-224
[14]Chen, Z. Q., Ma, Z. M. and Rockner, M., Quasi-homeomorphisms of Dirichlet forms [J]. Nagoya J.Math.136(1994):1-15
[15]Chen, Z. Q., Fitzsimmons, P. J., Kuwae, K. and Zhang, T. S., Perturbation of sym-metric Markov Processes [J]. Prob. Theory Relat. Fields.140,239-275, (2008)
[16]Chen, Z. Q., Fitzsimmons, P. J., Kuwae, K. and Zhang, T. S., On general perturbations of symmetric Markov processes [J]. J. Math. Pures et Appliquees 92(2009):363-374
[17]Chen, Z. Q., Fitzsimmons, P. J., Takeda, M., Ying, J. and Zhang, T. S., Absolute continuty of Symmetric Markov Processes [J]. Ann. Prob. Vol.32. No.3A.(2004):2067-2098
[18]Chen, Z. Q., Fukushima, M. and Ying, J., Trace of symmetric Markov processes and their Characterizations [J]. Ann. Prob. V.34, No.3(2006):1052-1102
[19]Chen, Z. Q., Fukushima, M. and Ying, J., Entrance law, Exit system and Levy System of time changed processes[J]. Illinois Journal of Mathematics, V50, N2, Summer 2006, 269-312
[20]Chen, Z. Q. and Fukushima, M., Symmetric Markov Processes, Time Change and Boundary Theory. (2009) Book manuscript
[21]Chen, Z. Q. and Zhang, T. S., Girsanov and Feynman-Kac Type Transformations for Symmetric Markov Processes [J]. Ann.I.Poincare-PR38,4(2002):475-505
[22]Chen, Z. Q, Zhang, T. S., Time-reversal and elliptic boundary value problems [J]. Ann. Probab.37(2009):1008-1043
[23]Chen, Z. Q., On Feynman-Kac perturbation of symmetric Markov processes [J]. In Proceedings of Functional Analysis IX, Dubrovnik, Croatia,2005:39-43
[24]Chung, K. L., Lectures from Markov Processes to Brownian Motion [M]. Berlin Hei-delberg New York Tokyo, Springer,1982
[25]Chung, K. L, and Walsh, J. B., Markov Processes, Brownian Motion, and Time sym-metry [M]. Berlin New York Tokyo, Springer,2006
[26]Chung, K. L. and Zhao, Z. X., From Brownian Motion to Schrodinger's Equation [M]. Berlin Heidelberg, Springer,1995
[27]Conway, J. B., A course in Functional Analysis [M]. World Publishing corporation, BeiJing,2003
[28]Dong, Z., New results on Dirichlet Forms and constructions of General Markov pro-cesses [D]. Dessertation, AMSS,CAS, Beijing,1996
[29]Dong, Z., construction of Markov process for Sub-Markov Resolvents on Lp(E,m) [J]. Sci. Chin.(A) Vol.40(9) (1997):897-908
[30]Doob, J. L., conditional Brownian Motion And The Boundary Limits of Harmonic Functions [J]. Bull. Soc. Math. France,85(1957):431-458
[31]Doob, J. L., Classical Potential Theory and its probabilistic counterpart [M]. Springer-Verlag New York,1983
[32]Durrett, R., Brownian motion and Martingales in Analysis [M]. Wadsworth Advanced Books & Software, California,1984
[33]Fitzsimmons, P. J., On the quasi-regularity of semi-Dirichlet forms [J]. Potential Anal. 15(2001):158-185
[34]Fitzsimmons, P. J. and Getoor, R. K., Levy system and time changes [J]. PrePrint.
[35]Fitzsimmons, P. J. and Kuwae, K., Nonsymmetric perturbations of symmetric Dirichlet forms [J]. J.Funct. Anal.208(2004):140-162
[36]Some problems of conditional Markov processes [J]. Workshop on stochastic and har-monic analysis of processes with jumps, Angers, Franc,2-9, may,2006
[37]Fukushima, M., Almost polar sets and ergodic theorem [J]. J. Math. Soc. Japan. V.26(1)(1974):17-32
[38]Fukushima, M., Oshima, Y. and Takeda, M., Dirichlet Forms and Symmetric Markov Processes [M]. Walter de Gruyrer, Berlin,1994
[39]Gerhard, W. D., The probabilistic of the Dirichlet Problem for 1/2∞++b with singular coefficents [J]. J. Theor. Prob.5(1992):503-520
[40]Glover, J., Rao, M., Hunt's Hypothesis and Getoor conjectue [J]. Ann. Prob. 14(3)(1987):1085-1087
[41]Glover, J., Rao, M., sikic, H. and Song, R., Quadratic Forms corresponding to the Generalized Schrodinger Semigroups [J]. J. Funct. Anal.125(1994):358-378
[42]龚光鲁.随机微分方程引论[M].北京,北京大学出版社第二版
[43]韩新方,马丽.非对称狄氏型的扰动及其对应的无穷小生成元[J].海南师范大学学报(自然科学版).20(1)(2007)：13-18
[44]Hu, Z. C.,Ma, Z. M. and Sun, W., Extensions of Levy-Khintchine formula and Beurling-Deny formula in semi-Dirichlet forms setting [J]. J. Funct. Anal.239(2006): 179-213
[45]Jacob, N. and Schilling, R.L., Towards an Lp Potential Theory for Sub-Markovian Semigroups:Kernels and Capacities [J]. Acta Mathematica Sinica, English Series Vol.22(4)(2006):1227-1250
[46]Klebaner, F. C., Introduction to stochastic calculus with applications [M]. Imperial college press,1998
[47]Kuwae, K., Maximum principles for subharmonic functions via local semi-Dirichlet forms [J]. Canad. J. Math.60(2008):822-874
[48]Kanda, M., Two theorems on capacity for Markov processes with stationary indepen-dent increments [J]. Z.Wahrscheinlichkeitstheorie verw. Gebiete 35(1976):159-165
[49]Kanda, M., Characterization of Semipolar Sets for Proceses with Stationary indepen-dent increments [J]. Z.Wahrscheinlichkeitstheorie verw.Gebiete 42(1978):141-154
[50]Kanda, M., A note on capacitary measures of semipolar sets [J]. Ann. Prob. 17(1)(1989):379-384
[51]Kim, P. and Song, R. M., Two-sided estimates on the density of Brownian motion with singular drift [J]. Illionis J.M, V.50, No.3(2006):635-688
[52]Ma, Z. M., Overbeck, L. and Rockner, M., Markov processes associated with Semi-Dirichlet forms [M]. Osaka J. Math,32(1995):97-119
[53]Ma, Z. M. and Rockner, M., Introduction to the Theory of (Non-Symmetric) Dirichlet Forms [M]. Springer-Verlag, Berlin,1992
[54]Ma, Z. M. and Rockner, M., Markov Processes Associted With Positivity preserving coercive Forms [J]. Can. J. Math. V.47(4)(1995):817-840
[55]Ma, Z M. and Sun, W., Some topics on Dirichlet forms [J]. New Trend in Stochastic Analysis and Related Topics, a Volume in Honour of Professor K.D. Elworthy, (2010)
[56]Ma, L. and Sun, W., On the generalized Feynman-Kac transformation for nearly sym-metric Markov processes [J]. J. Theor. Prob., (2010)
[57]Y. Oshima., The non-symmetric Dirichlet form [M]. Erlangen,1988
[58]Oshima, M., On the exceptionality of some semipolar sets of time inhomogeneous markov processes [J]. Tohoku Math. J.54(2002):443-449
[59]Pazy, A., Semigroups of linear operators and Applications to partial differential equa-tions [M]. New York, Springer-Verlag,1983
[60]钱敏平.随机过程论[M]。北京,北京大学出版社,1997。
[61]Rao, M., Hunt's Hypothesis for levy processes [J]. Proceedings of the American Math-ematical Society,104(2) 1988:621-624
[62]Rao, M., J. L. Doob [J]. Ann. Prob., V.37(5) 2009:1664-1670
[63]Sharpe, M., General Theory of Markov Processes [M]. Boston San Diego New York Tokyo, Academic Press, INC.1988
[64]Silverstein, L., The sector condition implies that semipolar sets are quasi-polar [J]. Z. Wahrscheinlichkeitstheorie verw. Gebiete.41(1977):13-33
[65]Servet Martinez and Jaime San Martin, Rates of Decay and h-Processes for One Di-mensional Diffusions conditioned on Non-Absorption [J]. Journal of Theoretical Prob. Vol.14, No.1(2001)
[66]Stratonovich, R. L., Conditional Markov Processes [J]. Theory Probab. Appl. Volume 5, Issue 2, (January 1960):156-178
[67]Stannat, W., Generalized Dirichlet forms and Markov processes [J]. C.R. Acad. Sci. Paris.319(1994):1063-1068
[68]Stannat, W., The theory of generalized Dirichler foms and its applications in analysis and stochastics [D]. Mem. Amer. Math. Soc.142 (1999):678
[69]Takeda, M., Asymptotic properties of generalized Feynman-Kac functionals [J]. J. Funct. Anal.9 (1998):261-291
[70]Takeda, M., Topics on Dirichlet Forms and Symmetric Markov Processes [J]. SUGAKU EXPOSITION Vol.12(2)(1999):201-221
[71]Takeda, M. and Zhang, T. S., Asymptotic Properties of Additive Functionals of Brow-nian Motion [J]. Ann. Prob.,25(2)(1997):940-952
[72]Wang, Y. X., Transformation of Dirichlet form [D]. [Ph.D.Dissertation] Institute of Applied Mathematices Academia Sinica,1994
[73]严加安,彭实戈,方诗赞等,随机分析选讲[M].科学出版社,北京,2000
[74]Ying, J. G., Hunt-假设与Getoor猜想.Homepage of Jiangang Ying, Document of Doc
[75]Ying, J. G., Remarks on h-transform and Drift [J]. Chin. Ann. Math.19B(4)1998: 473-478
[76]Jin, M. W. and Ying, J. G., Additive functionals and Perturbation of semigroup [J]. Chin. Ann. Math.22B(4)(2001):503-512
[77]Yoshida, K., Functional Analysis [M]. Beijing World Publishing corporation,1999
[78]Zhang, T. Z., Generalized Feynman-Kac Semigroups, Associated Quadratic Forms and Asymptotic Properties [J]. Potential Anal.14(2001):387-408
[79]张恭庆,郭懋正.泛函分析讲义[M].北京,北京大学出版社,1990
[80]Zhao, Z. X., Subcriticality, positivity and Gaugeability of the Schrodinger operator [J]. Bulletin. Amer. Math. Soci.23(2)(1990):513-517
[81]Zhao, Z. X., Schrodinger conditional brownian motion and stochastic calculus of vari-ations [J]. An International Journal of Probability and Stochastic Processes, Vol.18, Issue 1 July 1986:1-15
[2]Albeverio, S. and Ma, Z. M., Additive Functional, Nowhere Radon and Kato Class Smooth Measures Associated With Dirichlet Forms [J]. Osaka J.Math.29(1992):247-265
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[7]Blumenthal, R. M. and Getoor, R. K., Markov processes and Potential Theory [M]. Academic Press, New York,1968
[8]Chen, C. Z., Ma, Z. M. and Sun, W., On Girsanov and generalized Feynman-Kac Transformations for symmetric Markov processes [J]. Infinite Dimensional Analysis, Quantum Probability and Related Topics, V.10, No.2(2007):141-163
[9]陈传钟,孙玮.狄氏型的扰动及其结合的马氏过程[J].数学物理学报21A(2)(2001)：154-161
[10]Chen, C. Z. and Sun, W., Strong continuity of Generalized Feynman-Kac Semigroups: Necessary and Sufficient conditions [J]. J. Funct. Anal.237(2006):446-465
[11]陈传钟.广义狄氏型的扰动及其结合的马氏过程[J].应用数学学报24(4)(2001)：561-567
[12]Chen, C. Z.,狄氏型的扰动与Feynman-Kac半群[D].中南大学博士学位论文2004
[13]Chen,C. Z., A Note on Perturbation of Non-Symmetric Dirichlet Forms by Signed Smooth Measures [J]. Acta Mathematica Scientia.27B(1)(2007):119-224
[14]Chen, Z. Q., Ma, Z. M. and Rockner, M., Quasi-homeomorphisms of Dirichlet forms [J]. Nagoya J.Math.136(1994):1-15
[15]Chen, Z. Q., Fitzsimmons, P. J., Kuwae, K. and Zhang, T. S., Perturbation of sym-metric Markov Processes [J]. Prob. Theory Relat. Fields.140,239-275, (2008)
[16]Chen, Z. Q., Fitzsimmons, P. J., Kuwae, K. and Zhang, T. S., On general perturbations of symmetric Markov processes [J]. J. Math. Pures et Appliquees 92(2009):363-374
[17]Chen, Z. Q., Fitzsimmons, P. J., Takeda, M., Ying, J. and Zhang, T. S., Absolute continuty of Symmetric Markov Processes [J]. Ann. Prob. Vol.32. No.3A.(2004):2067-2098
[18]Chen, Z. Q., Fukushima, M. and Ying, J., Trace of symmetric Markov processes and their Characterizations [J]. Ann. Prob. V.34, No.3(2006):1052-1102
[19]Chen, Z. Q., Fukushima, M. and Ying, J., Entrance law, Exit system and Levy System of time changed processes[J]. Illinois Journal of Mathematics, V50, N2, Summer 2006, 269-312
[20]Chen, Z. Q. and Fukushima, M., Symmetric Markov Processes, Time Change and Boundary Theory. (2009) Book manuscript
[21]Chen, Z. Q. and Zhang, T. S., Girsanov and Feynman-Kac Type Transformations for Symmetric Markov Processes [J]. Ann.I.Poincare-PR38,4(2002):475-505
[22]Chen, Z. Q, Zhang, T. S., Time-reversal and elliptic boundary value problems [J]. Ann. Probab.37(2009):1008-1043
[23]Chen, Z. Q., On Feynman-Kac perturbation of symmetric Markov processes [J]. In Proceedings of Functional Analysis IX, Dubrovnik, Croatia,2005:39-43
[24]Chung, K. L., Lectures from Markov Processes to Brownian Motion [M]. Berlin Hei-delberg New York Tokyo, Springer,1982
[25]Chung, K. L, and Walsh, J. B., Markov Processes, Brownian Motion, and Time sym-metry [M]. Berlin New York Tokyo, Springer,2006
[26]Chung, K. L. and Zhao, Z. X., From Brownian Motion to Schrodinger's Equation [M]. Berlin Heidelberg, Springer,1995
[27]Conway, J. B., A course in Functional Analysis [M]. World Publishing corporation, BeiJing,2003
[28]Dong, Z., New results on Dirichlet Forms and constructions of General Markov pro-cesses [D]. Dessertation, AMSS,CAS, Beijing,1996
[29]Dong, Z., construction of Markov process for Sub-Markov Resolvents on Lp(E,m) [J]. Sci. Chin.(A) Vol.40(9) (1997):897-908
[30]Doob, J. L., conditional Brownian Motion And The Boundary Limits of Harmonic Functions [J]. Bull. Soc. Math. France,85(1957):431-458
[31]Doob, J. L., Classical Potential Theory and its probabilistic counterpart [M]. Springer-Verlag New York,1983
[32]Durrett, R., Brownian motion and Martingales in Analysis [M]. Wadsworth Advanced Books & Software, California,1984
[33]Fitzsimmons, P. J., On the quasi-regularity of semi-Dirichlet forms [J]. Potential Anal. 15(2001):158-185
[34]Fitzsimmons, P. J. and Getoor, R. K., Levy system and time changes [J]. PrePrint.
[35]Fitzsimmons, P. J. and Kuwae, K., Nonsymmetric perturbations of symmetric Dirichlet forms [J]. J.Funct. Anal.208(2004):140-162
[36]Some problems of conditional Markov processes [J]. Workshop on stochastic and har-monic analysis of processes with jumps, Angers, Franc,2-9, may,2006
[37]Fukushima, M., Almost polar sets and ergodic theorem [J]. J. Math. Soc. Japan. V.26(1)(1974):17-32
[38]Fukushima, M., Oshima, Y. and Takeda, M., Dirichlet Forms and Symmetric Markov Processes [M]. Walter de Gruyrer, Berlin,1994
[39]Gerhard, W. D., The probabilistic of the Dirichlet Problem for 1/2∞+
[40]Glover, J., Rao, M., Hunt's Hypothesis and Getoor conjectue [J]. Ann. Prob. 14(3)(1987):1085-1087
[41]Glover, J., Rao, M., sikic, H. and Song, R., Quadratic Forms corresponding to the Generalized Schrodinger Semigroups [J]. J. Funct. Anal.125(1994):358-378
[42]龚光鲁.随机微分方程引论[M].北京,北京大学出版社第二版
[43]韩新方,马丽.非对称狄氏型的扰动及其对应的无穷小生成元[J].海南师范大学学报(自然科学版).20(1)(2007)：13-18
[44]Hu, Z. C.,Ma, Z. M. and Sun, W., Extensions of Levy-Khintchine formula and Beurling-Deny formula in semi-Dirichlet forms setting [J]. J. Funct. Anal.239(2006): 179-213
[45]Jacob, N. and Schilling, R.L., Towards an Lp Potential Theory for Sub-Markovian Semigroups:Kernels and Capacities [J]. Acta Mathematica Sinica, English Series Vol.22(4)(2006):1227-1250
[46]Klebaner, F. C., Introduction to stochastic calculus with applications [M]. Imperial college press,1998
[47]Kuwae, K., Maximum principles for subharmonic functions via local semi-Dirichlet forms [J]. Canad. J. Math.60(2008):822-874
[48]Kanda, M., Two theorems on capacity for Markov processes with stationary indepen-dent increments [J]. Z.Wahrscheinlichkeitstheorie verw. Gebiete 35(1976):159-165
[49]Kanda, M., Characterization of Semipolar Sets for Proceses with Stationary indepen-dent increments [J]. Z.Wahrscheinlichkeitstheorie verw.Gebiete 42(1978):141-154
[50]Kanda, M., A note on capacitary measures of semipolar sets [J]. Ann. Prob. 17(1)(1989):379-384
[51]Kim, P. and Song, R. M., Two-sided estimates on the density of Brownian motion with singular drift [J]. Illionis J.M, V.50, No.3(2006):635-688
[52]Ma, Z. M., Overbeck, L. and Rockner, M., Markov processes associated with Semi-Dirichlet forms [M]. Osaka J. Math,32(1995):97-119
[53]Ma, Z. M. and Rockner, M., Introduction to the Theory of (Non-Symmetric) Dirichlet Forms [M]. Springer-Verlag, Berlin,1992
[54]Ma, Z. M. and Rockner, M., Markov Processes Associted With Positivity preserving coercive Forms [J]. Can. J. Math. V.47(4)(1995):817-840
[55]Ma, Z M. and Sun, W., Some topics on Dirichlet forms [J]. New Trend in Stochastic Analysis and Related Topics, a Volume in Honour of Professor K.D. Elworthy, (2010)
[56]Ma, L. and Sun, W., On the generalized Feynman-Kac transformation for nearly sym-metric Markov processes [J]. J. Theor. Prob., (2010)
[57]Y. Oshima., The non-symmetric Dirichlet form [M]. Erlangen,1988
[58]Oshima, M., On the exceptionality of some semipolar sets of time inhomogeneous markov processes [J]. Tohoku Math. J.54(2002):443-449
[59]Pazy, A., Semigroups of linear operators and Applications to partial differential equa-tions [M]. New York, Springer-Verlag,1983
[60]钱敏平.随机过程论[M]。北京,北京大学出版社,1997。
[61]Rao, M., Hunt's Hypothesis for levy processes [J]. Proceedings of the American Math-ematical Society,104(2) 1988:621-624
[62]Rao, M., J. L. Doob [J]. Ann. Prob., V.37(5) 2009:1664-1670
[63]Sharpe, M., General Theory of Markov Processes [M]. Boston San Diego New York Tokyo, Academic Press, INC.1988
[64]Silverstein, L., The sector condition implies that semipolar sets are quasi-polar [J]. Z. Wahrscheinlichkeitstheorie verw. Gebiete.41(1977):13-33
[65]Servet Martinez and Jaime San Martin, Rates of Decay and h-Processes for One Di-mensional Diffusions conditioned on Non-Absorption [J]. Journal of Theoretical Prob. Vol.14, No.1(2001)
[66]Stratonovich, R. L., Conditional Markov Processes [J]. Theory Probab. Appl. Volume 5, Issue 2, (January 1960):156-178
[67]Stannat, W., Generalized Dirichlet forms and Markov processes [J]. C.R. Acad. Sci. Paris.319(1994):1063-1068
[68]Stannat, W., The theory of generalized Dirichler foms and its applications in analysis and stochastics [D]. Mem. Amer. Math. Soc.142 (1999):678
[69]Takeda, M., Asymptotic properties of generalized Feynman-Kac functionals [J]. J. Funct. Anal.9 (1998):261-291
[70]Takeda, M., Topics on Dirichlet Forms and Symmetric Markov Processes [J]. SUGAKU EXPOSITION Vol.12(2)(1999):201-221
[71]Takeda, M. and Zhang, T. S., Asymptotic Properties of Additive Functionals of Brow-nian Motion [J]. Ann. Prob.,25(2)(1997):940-952
[72]Wang, Y. X., Transformation of Dirichlet form [D]. [Ph.D.Dissertation] Institute of Applied Mathematices Academia Sinica,1994
[73]严加安,彭实戈,方诗赞等,随机分析选讲[M].科学出版社,北京,2000
[74]Ying, J. G., Hunt-假设与Getoor猜想.Homepage of Jiangang Ying, Document of Doc
[75]Ying, J. G., Remarks on h-transform and Drift [J]. Chin. Ann. Math.19B(4)1998: 473-478
[76]Jin, M. W. and Ying, J. G., Additive functionals and Perturbation of semigroup [J]. Chin. Ann. Math.22B(4)(2001):503-512
[77]Yoshida, K., Functional Analysis [M]. Beijing World Publishing corporation,1999
[78]Zhang, T. Z., Generalized Feynman-Kac Semigroups, Associated Quadratic Forms and Asymptotic Properties [J]. Potential Anal.14(2001):387-408
[79]张恭庆,郭懋正.泛函分析讲义[M].北京,北京大学出版社,1990
[80]Zhao, Z. X., Subcriticality, positivity and Gaugeability of the Schrodinger operator [J]. Bulletin. Amer. Math. Soci.23(2)(1990):513-517
[81]Zhao, Z. X., Schrodinger conditional brownian motion and stochastic calculus of vari-ations [J]. An International Journal of Probability and Stochastic Processes, Vol.18, Issue 1 July 1986:1-15