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. (C) 2010 Elsevier B.V. All rights reserved.