Random walks colliding before getting trapped

Abstract

Let P be the transition matrix of a finite, irreducible and reversible Markov chain.We say the continuous time Markov chain X has transition matrix P and speed λ if it jumps at rate λ according to the matrix P.Fix λX; λY; λZ λ ≥ 0, then let X; Y and Z be independent Markov chains with transition matrix P and speeds λX; λY and λZ respectively, all started from the stationary distribution.What is the chance that X and Y meet before either of them collides with Z? For each choice of λX; λY and λZ with max(λx; λy) > 0, we prove a lower bound for this probability which is uniform over all transitive, irreducible and reversible chains.In the case that λX = λY = 1 and λZ = 0 we prove a strengthening of our main theorem using a martingale argument.We provide an example showing the transitivity assumption cannot be removed for general λX; λY and λZ.