From one dimensional diffusions to symmetric Markov processes

For an absorbing diffusion X0X0 on a one dimensional regular interval II with no killing inside, the Dirichlet form of X0X0 on L2(I;m)L2(I;m) and its extended Dirichlet space are identified in terms of the canonical scale ss of X0X0, where mm is the canonical measure of X0X0. All possible symmetric extensions of X0X0 will then be considered in relation to the active reflected Dirichlet space of X0X0. 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 Rd,d≥3, possessing two branches of infinite cones.