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].
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