Intrinsic metric and one-dimensional diffusions

One-dimensional strongly local and regular Dirichlet forms which are irreducible can always be characterized by the so-called scale function s and speed measure m . In this paper we derive the intrinsic metric of such a Dirichlet form in terms of s and m . As an application, we give a new characterization of regular Dirichlet subspaces and extensions of Brownian motion, comparing to Fang et al. (2005) and Li and Ying (2019). Finally two examples on volume doubling property of regular Dirichlet subspaces of Brownian motion are presented.