Abstract
We give a simple proof of the symmetry of a minimal diffusion X0 on a one-dimensional open interval I with respect to the attached canonical measure m along with the identification of the Dirichlet form of X0 on L2(I; m) in terms of the triplet (s,m,k) attached to X0. The L2-generators of X0 and its reflecting extension Xr are then readily described. We next use the associated reproducing kernels in connecting the L2-setting to the traditional Cb-setting and thereby deduce characterizations of the domains of Cb-generators of X0 and Xr by means of boundary conditions. We finally identify the Cb-generators for all other possible symmetric diffusion extensions of X0 and construct by that means all diffusion extensions of X0 in [IM2].
