Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gaskets

We consider a class of random recursive Sierpinski gaskets and examine the short-time asymptotics of the on-diagonal transition density for a natural Brownian motion. In contrast to the case of divergence form operators in Rn or regular fractals we show that there are unbounded fluctuations in the leading order term. Using the resolvent density we are able to explicitly describe the fluctuations in time at typical points in the fractal and, by considering the supremum and infimum of the on-diagonal transition density over all points in the fractal, we can also describe the fluctuations in space.