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

2014/12/05 by Liping Li, Li, Liping, Jiangang Ying +1
Mathematics · #31C25 #60J55 #60J60 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:31C25 #msc:60J55 #msc:60J60

paper · pdf · doi:10.48550/arxiv.1412.1896

arxiv created 2016/05/04 · arxiv updated 2016/05/05

Abstract

The main purpose of this paper is to explore the structure of regular subspaces of 1-dim Brownian motion. As outlined in \citeFMG every such regular 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 subspace is nothing but Brownian motion. Their traces on F still inherit the inclusion relation, in other words, the trace Dirichlet form of regular subspace on F is still a regular subspace of trace Dirichlet form of one-dimensional Brownian motion on F. Moreover we have proved that the trace of Brownian motion on F may be decomposed into two part, one is the trace of the regular subspace on F, which has only the non-local part and the other comes from the orthogonal complement of the regular subspace, which has only the local part. Actually the orthogonal complement of regular subspace corresponds to a time-changed Brownian motion after a darning transform.

Related

• Regular Dirichlet extensions of one-dimensional Brownian motion
• Weak convergence of regular Dirichlet subspaces
• Regular subspaces of skew product diffusions
• The orthogonal complements of H¹(ℝ) in its regular Dirichlet extensions
• The first passage time density of Brownian motion and the heat equation with Dirichlet boundary condition in time dependent domains