Skew Brownian motion-type of extensions

We consider a class of possible extensions of a given symmetric Feller process Z, from R\{0} to the entire real line R, depending on a parameter alpha is an element of [0, 1]. It is proved that the proposed extension exists if alpha = 1/2; for alpha not equal 1/2, exists if and only if Z(t) does not jump over 0 (e.g., if Z(t) is a diffusion).