Probability density for diffusion on fractals

Abstract

We propose an asymptotic formula for the probability density $P_F\mathrm{}(x,t)\mathrm{}$ that a random walker starts at an arbitrary origin on a fractal at time 0 and arrives at site $x$ at time $t$. We have numerically verified the proposed form for $P_F$ by testing a composition rule for probabilities which requires that the walker must have been on some site of the fractal at any intermediate time. We further write a Wiener integral representation of $P_F$.