B.M. Hambly and O.D. Jones,
Asymptotically one-dimensional diffusion on the Sierpinski gasket 
and multi-type branching processes with varying environment.

J. Theoret. Prob., 15(2002), 285-322.

Abstract:
Asymptotically one-dimensional diffusions on the Sierpinski gasket
constitute a one parameter family of processes with significantly
different behaviour to the Brownian motion.  Due to homogenization
effects they behave globally like the Brownian motion, yet locally
they have a preferred direction of motion. We calculate the spectral
dimension for these processes and obtain short time heat kernel
estimates in the Euclidean metric. The results are derived using
branching process techniques, and we give estimates for the left tail
of the limiting distribution for a supercritical multi-type branching
process with varying environment.