From one dimensional diffusions to symmetric Markov processes

For an absorbing diffusion X 0 on a one dimensional regular interval I with no killing inside, the Dirichlet form of X 0 on L 2 ( I ; m ) and its extended Dirichlet space are identified in terms of the canonical scale s of X 0 , where m is the canonical measure of X 0 . All possible symmetric extensions of X 0 will then be considered in relation to the active reflected Dirichlet space of X 0 . 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 R d , d ≥ 3 , possessing two branches of infinite cones.