The Last Passage Problem on Graphs

We consider a Brownian motion on a general graph, that starts at time t=0 from some vertex O and stops at time t somewhere on the graph. Denoting by g the last time when O is reached, we establish a simple expression for the Laplace Transform, L, of the probability density of g. We discuss this result for some special graphs like star, ring, tree or square lattice. Finally, we show that L can also be expressed in terms of primitive orbits when, for any vertex, all the exit probabilities are equal.