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. Around the years 1930, P Levy [1] got several arc-sine laws concerning the 1D Brownian Motion on an infinite line. Let us consider such a process starting at t = 0 from the origin O and stopping at time t and denote by g the last time when O is reached. The second arc-sine law discovered by Levy concerns the probability law of g. It can be stated as follows: P (g < u) = 2 π arcsin √ u t (1) with the probability density Pt(u) ≡ dP (g < u) du = 1 π 1 √ u(t− u) (2) In particular, the Laplace transform of Pt(u) is written: ∫ ∞ 0 dt e−γt ∫ t 0 du Pt(u) e−ξu = 1 √ γ(γ + ξ) (3) Levy also proved that the same law occurs when g represents the time spent in the region (x > 0) (first arc-sine law). The infinite line with only one point, O, specified, can be viewed as a kind of very simple graph consisting in one vertex, O, and two semi-infinite lines originating from O. Our goal in this Letter is to get an analogue of the second arc-sine law (3) but for a quite general graph.