The Last Passage Problem on Graphs

TL;DR

This paper derives a simple Laplace Transform expression for the distribution of the last time a Brownian motion on a graph visits its starting vertex, with applications to various graph structures and orbit-based representations.

Contribution

It introduces a novel, simple formula for the Laplace Transform of the last passage time on graphs, extending to special cases and orbit-based expressions.

Findings

Derived explicit Laplace Transform for last passage time

Applied results to star, ring, tree, and lattice graphs

Connected the transform to primitive orbit sums under symmetric exit probabilities

Abstract

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.