A characterization of certain diffusions on the Sierpinski gasket by the exit distributions

LetK be the Sierpinski gasket with verticesa ₁,a ₂,a ₃ forming an equilateral triangle. Suppose that (X _t ,P ^x ) and (Y _t ,Q ^x ) are diffusions onK with the same hitting probabilities to verticesa ₁,a ₂,a ₃. We show that ifX is an asymptotically one-dimensional diffusion or ap-stream diffusion, thenY is a time-change ofX.Part of this work was done while the author was an Alexander von Humboldt fellow at the Universität des Saarlandes in Saarbrücken, Germany.