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

LetK be the Sierpinski gasket with verticesa₁,a₂,a₃ forming an equilateral triangle. Suppose that (X_t,P^x) and (Y_t,Q^x) are diffusions onK with the same hitting probabilities to verticesa₁,a₂,a₃. 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.