Heat Kernel Estimates and Homogenization for Asymptotically Lower Dimensional Processes on Some Nested Fractals

We consider the class of diffusions on fractals first constructed in [12] on the Sierpinski and abc gaskets. We give an alternative construction of the diffusion process using Dirichlet forms and extend the class of fractals considered to some nested fractals. We use the Dirichlet form to deduce Nash inequalities which give upper bounds on the short and long time behaviour of the transition density of the diffusion process. For short times, even though the diffusion lives mainly on a lower dimensional subset of the fractal, the heat flows slowly. For the long time scales the diffusion has a homogenization property in that rescalings converge to the Brownian motion on the fractal.