The differential equation describing random walks on the Koch curve

Consider a particle which is released at some point on a fractal and which moves about the fractal at random. A long standing goal has been to determi ne a differential equation governing the probability density function which describes this walk. As well as being interesting in its own right, this p roblem is thought to provide an insight into the problem of anomalous diffu sion. Many attempts to derive such an equation have been made, all with lim ited success., perhaps because of the tension between smoothness required b y differential equation tools and the lack of smoothness inherent in fracta ls. Here we present, for the first time, the equation governing the random walk on a simple fractal-the Koch curve. We show that this equation makes c omputation of the probability density function for this problem a simple ma tter.