From one dimensional diffusions to symmetric Markov processes

For an absorbing diffusion X0 on a one dimensional regular interval I with no killing inside, the Dirichlet form of X0 on L2(I;m) and its extended Dirichlet space are identified in terms of the canonical scale s of X0, where m is the canonical measure of X0. All possible symmetric extensions of X0 will then be considered in relation to the active reflected Dirichlet space of X0. Furthermore quite analogous considerations will be made for possible symmetric extensions of a specific diffusion in a higher dimension, namely, a time changed transient reflecting Brownian motion on a closed domain of , possessing two branches of infinite cones.