正则Dirichlet形式的正则子空间
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摘要
正则Dirichlet形式的正则子空间问题,即存在性及其刻画等问题,是作者关注近20年的问题,该问题来自第二作者对于Markov过程的Killing变换的Dirichlet形式刻画问题的研究,这个问题是Dirichlet形式理论的一个基本问题,在最近10年中取得了一点进展.本文将主要叙述问题及其背景,并介绍围绕该问题得到的一些结果与遗留的问题.
This paper focuses on the problem of the existence and representation of regular Dirichlet subspaces of a regular Dirichlet form, which was raised by the second author when studying the characterization of Killing transform of Markov processes by Dirichlet forms, and has been studied for about 20 years. A brief survey on recent progress on this problem in the frame of 1-dimensional symmetric di?usions is presented.
This paper focuses on the problem of the existence and representation of regular Dirichlet subspaces of a regular Dirichlet form, which was raised by the second author when studying the characterization of Killing transform of Markov processes by Dirichlet forms, and has been studied for about 20 years. A brief survey on recent progress on this problem in the frame of 1-dimensional symmetric di?usions is presented.
引文
1 Fang X,Fukushima M,Ying J.On regular Dirichlet subspaces of H1(I)and associated linear diffusions.Osaka JMath,2005,42:27-41
2 Li L,Ying J.Regular subspaces of Dirichlet forms.In:Festschrift Masatoshi Fukushima.Hackensack:World Scientific Publishing,2015,397-420
3 Li L,Ying J.On structure of regular Dirichlet subspaces for one-dimensional Brownian motion.Ann Probab,2017,45:2631-2654
4 Fang X,He P,Ying J.Dirichlet forms associated with linear diffusions.Chin Ann Math Ser B,2010,31:507-518
5 Li L,Ying J.On symmetric linear diffusions.Trans Amer Math Soc,2017,in press
6 Li L,Ying J.Regular Dirichlet extensions of one-dimensional Brownian motion.ArXiv:1606.00630,2016
7 Blumenthal R M,Getoor R K.Markov Processes and Potential Theory.New York:Academic Press,1968
8 Fukushima M,Oshima Y,Takeda M.Dirichlet Forms and Symmetric Markov Processes.Berlin:Walter de Gruyter,2011
9 Ma Z M,R¨ockner M.Introduction to the Theory of(Nonsymmetric)Dirichlet Forms.Berlin-Heidelberg:SpringerVerlag,1992
10 Ying J.Bivariate Revuz measures and the Feynman-Kac formula.Ann Inst H Poincar′e Probab Statist,1996,32:251-287
11 Ying J.Killing and subordination.Proc Amer Math Soc,1996,124:2215-2223
12 Fukushima M,Ying J.A note on regular Dirichlet subspaces.Proc Amer Math Soc,2003,131:1607-1611
13 Chen Z Q,Fukushima M,Ying J.Traces of symmetric Markov processes and their characterizations.Ann Probab,2006,34:1052-1102
14 Fukushima M,He P,Ying J.Time changes of symmetric diffusions and Feller measures.Ann Probab,2004,32:3138-3166
15 Fukushima M.Dirichlet Forms and Markov Processes.Amsterdam-Oxford-New York:Kodansha and North-Holland,1980
16 Fukushima M.From one dimensional diffusions to symmetric Markov processes.Stochastic Process Appl,2010,120:590-604
17 Ying J,Zhao M.The uniqueness of symmetrizing measure of Markov processes.Proc Amer Math Soc,2010,138:2181-2185
18 Fitzsimmons P J,Li L.Class of smooth functions in Dirichlet spaces.ArXiv:1611.06778,2017
19 Ito K,McKean H P Jr.Diffusion Processes and Their Sample Paths.Berlin-New York:Springer-Verlag,1974
2 Li L,Ying J.Regular subspaces of Dirichlet forms.In:Festschrift Masatoshi Fukushima.Hackensack:World Scientific Publishing,2015,397-420
3 Li L,Ying J.On structure of regular Dirichlet subspaces for one-dimensional Brownian motion.Ann Probab,2017,45:2631-2654
4 Fang X,He P,Ying J.Dirichlet forms associated with linear diffusions.Chin Ann Math Ser B,2010,31:507-518
5 Li L,Ying J.On symmetric linear diffusions.Trans Amer Math Soc,2017,in press
6 Li L,Ying J.Regular Dirichlet extensions of one-dimensional Brownian motion.ArXiv:1606.00630,2016
7 Blumenthal R M,Getoor R K.Markov Processes and Potential Theory.New York:Academic Press,1968
8 Fukushima M,Oshima Y,Takeda M.Dirichlet Forms and Symmetric Markov Processes.Berlin:Walter de Gruyter,2011
9 Ma Z M,R¨ockner M.Introduction to the Theory of(Nonsymmetric)Dirichlet Forms.Berlin-Heidelberg:SpringerVerlag,1992
10 Ying J.Bivariate Revuz measures and the Feynman-Kac formula.Ann Inst H Poincar′e Probab Statist,1996,32:251-287
11 Ying J.Killing and subordination.Proc Amer Math Soc,1996,124:2215-2223
12 Fukushima M,Ying J.A note on regular Dirichlet subspaces.Proc Amer Math Soc,2003,131:1607-1611
13 Chen Z Q,Fukushima M,Ying J.Traces of symmetric Markov processes and their characterizations.Ann Probab,2006,34:1052-1102
14 Fukushima M,He P,Ying J.Time changes of symmetric diffusions and Feller measures.Ann Probab,2004,32:3138-3166
15 Fukushima M.Dirichlet Forms and Markov Processes.Amsterdam-Oxford-New York:Kodansha and North-Holland,1980
16 Fukushima M.From one dimensional diffusions to symmetric Markov processes.Stochastic Process Appl,2010,120:590-604
17 Ying J,Zhao M.The uniqueness of symmetrizing measure of Markov processes.Proc Amer Math Soc,2010,138:2181-2185
18 Fitzsimmons P J,Li L.Class of smooth functions in Dirichlet spaces.ArXiv:1611.06778,2017
19 Ito K,McKean H P Jr.Diffusion Processes and Their Sample Paths.Berlin-New York:Springer-Verlag,1974