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July 2017 On structure of regular Dirichlet subspaces for one-dimensional Brownian motion
Liping Li, Jiangang Ying
Ann. Probab. 45(4): 2631-2654 (July 2017). DOI: 10.1214/16-AOP1121


The main purpose of this paper is to explore the structure of regular Dirichlet subspaces of one-dimensional Brownian motion. As stated in [Osaka J. Math. 42 (2005) 27–41], every such regular Dirichlet subspace can be characterized by a measure-dense set $G$. When $G$ is open, $F=G^{c}$ is the boundary of $G$ and, before leaving $G$, the diffusion associated with the regular Dirichlet subspace is nothing but Brownian motion. Their traces on $F$ still inherit the inclusion relation, in other words, the trace Dirichlet form of regular Dirichlet subspace on $F$ is still a regular Dirichlet subspace of trace Dirichlet form of one-dimensional Brownian motion on $F$. Moreover, we shall prove that the trace of Brownian motion on $F$ may be decomposed into two parts; one is the trace of the regular Dirichlet subspace on $F$, which has only the nonlocal part and the other comes from the orthogonal complement of the regular Dirichlet subspace, which has only the local part. Actually the orthogonal complement of regular Dirichlet subspace corresponds to a time-changed absorbing Brownian motion after a darning transform.


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Liping Li. Jiangang Ying. "On structure of regular Dirichlet subspaces for one-dimensional Brownian motion." Ann. Probab. 45 (4) 2631 - 2654, July 2017.


Received: 1 February 2016; Revised: 1 April 2016; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 1376.31012
MathSciNet: MR3693971
Digital Object Identifier: 10.1214/16-AOP1121

Primary: 31C25, 60J55
Secondary: 60J60

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 4 • July 2017
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