On structure of regular Dirichlet subspaces for one-dimensional Brownian motion

Abstract

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.