保正半群的hh？-变换及其相关问题的研究

The theory of quasi-regular Dirichlet form and its applications are one of important research branches in the field of stochastic analysis. Based on the theory,Professor Ma Z.M. and his partners investigated the h-transform of Positivity preserving coercive closed form, proved that the necessary and sufficient condition for the Positivity preserving coercive closed form to be associated with a special standard Markov process is that the form is quasi-regular. As it is well-known that Positivity preserving form is associated with a pair of positivity preserving semigroups. Recently the applicant and his partners studied the transforms of positivity preserving semigroups and proved that the necessary and sufficient condition for positivity preserving coercive closed form to be associated with a pair of special standard Markov processes is that the form is quasi-regulary. Based on these researches, this project consists of two parts as follows: In the first part, we will discribe when the transformed quadratic forms satisfy sector condition and investigate the closability of the domain; In the second part, we will investigate the analytic representation of the new form and try to characterize the properties of the paths of the pair of special standard processes by properties of positivity preserving coercive closed form. This project will further enrich and develop the theory of quasi-regular Dirichlet forms and Markov processes, so it is of theoretical significance and application value.

拟正则狄氏型理论及其应用是现代随机分析领域的一个重要研究方向.借助于该理论,马志明院士等人研究了保正型的h-变换,证明了保正型（h-）结合一个特殊标准马氏过程的充要条件是保正型是拟正则的.众所周知,保正型联系的半群是保正半群.最近,申请人及其合作者研究了保正半群的hĥ-变换,证明了保正型（在一定意义下）结合一对特殊标准马氏过程的充要条件是该保正型是拟正则的等结果.在此基础上,本课题将研究如下问题：（1）刻画保正型（hĥ-）变换后所得新的二次型在什么情况下满足弱扇条件,并研究新二次型的表达式,定义域及其可闭性等问题.（2）给定保正型的表达式,研究新二次型的分析结构,进而利用保正型的性质刻画与之结合的对偶马氏过程的轨道连续性.本课题的研究将进一步丰富和完善拟正则狄氏型及马氏过程相关理论,有一定的理论意义和应用价值.

本项目主要研究保正型h\hat{h}-变换的相关问题，包括h\hat{h}-变换的理论及应用和狄氏型理论中一些经典结论和工具的推广。已基本上完成了项目的计划要求，执行情况良好。主要取得了如下成果：1. 将有界线性算子的积分表示理论用在保正半群的h\hat{h}-变换中，定义了保正半群的h\hat{h}-变换，证明了变换后的半群是对偶的强连续压缩次马氏半群，并且在拟正则的条件下，联系着一对对偶的马氏过程。我们从新的角度研究了保正半群的h\hat{h}-变换，这部分内容丰富了保正型框架下h\hat{h}-变换的相关结果。2.在半狄氏型框架下，定义了Kato-类光滑测度，证明了此Kato-类光滑测度与关于Green核的Kato-类等价，给出了Kato-类光滑测度可以逼近光滑测度，并且某种程度上受其所在状态空间上参考测度的控制。这一新的结果推广了对称狄氏型框架下的相应结果，为研究半狄氏型的扰动、保正型的h-变换、广义Feynman-Kac半群的强连续、大偏差等打下了基础。3. 在非对称狄氏型框架下，利用Riesz表示定理，定义了非对称狄氏型空间上的一种一一映射，构造了非对称狄氏型定义域中的元素和其对称型定义域中的元素之间的对应关系，并证明了该对应是线性并且有界的。该结果探索了对称狄氏型与非对称狄氏型的某种联系。4.研究了函数的可积性等，为寻找h-变换中使用的过分函数提供了素材。5.访问加拿大Concordia大学数学与统计系1人次，中科院数学与系统科学研究院应用数学所3人次。