Regular Dirichlet extensions of one-dimensional Brownian motion

2016/06/02 by Liping Li, Li, Liping, Jiangang Ying +1
Mathematics · #60J55 #FOS: Mathematics #Primary 31C25 #Probability (math.PR) #Secondary 60J60 #math.PR #msc:31C25 #msc:60J55 #msc:60J60

paper · pdf · doi:10.48550/arxiv.1606.00630

29 pages with 2 figures

arxiv created 2016/06/02 · arxiv updated 2016/06/03

Abstract

The regular Dirichlet extension is the dual concept of regular Dirichlet subspace. The main purpose of this paper is to characterize all the regular Dirichlet extensions of one-dimensional Brownian motion and to explore their structures. It is shown that every regular Dirichlet extension of one-dimensional Brownian motion may essentially decomposed into at most countable disjoint invariant intervals and an E-polar set relative to this regular Dirichlet extension. On each invariant interval the regular Dirichlet extension is characterized uniquely by a scale function in a given class. To explore the structure of regular Dirichlet extension we apply the idea introduced in [17], we formulate the trace Dirichlet forms and attain the darning process associated with the restriction to each invariant interval of the orthogonal complement of H¹_e(ℝ) in the extended Dirichlet space of the regular Dirichlet extension. As a result, we find an answer to a long-standing problem whether a pure jump Dirichlet form has proper regular Dirichlet subspaces.

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