Walshs Brownian Motion-Type of Extensions

Let (X t ) be a rotation invariant Feller process on the state space F (element of)R2{} consisting of finite number of rays, meeting at 0. We study a certain class of possible strong Markov extensions of (X t ) to F (n-ary union),{} given the corresponding radial extension to [0, (infinity)). A well-known example is the class of “Walshʹs Brownian motions,” in the case where (X t ) is the Brownian motion on F. It turns out that while the symmetric extension of “Walshʹs Brownian motion-type” always exists, the non-symmetric extension exists iff (X t ), roughly speaking, does not jump from one ray to another before hitting 0.