Martin boundaries associated with a killed random walk

Martin Boundaries and Random Walks

The first three sections give a quick overview of Martin boundary theory and state the main results. The succeeding sections will flesh out the details, and give proofs and examples. Virtually all of the results below are classical. The article Doob (1959) and the book by Kemeny, Snell, and Knapp (1976) are good sources for additional details. A recent survey article by Wolfgang Woess (1994) ha...

A Random Walk with Exponential Travel Times

Consider the random walk among N places with N(N - 1)/2 transports. We attach an exponential random variable Xij to each transport between places Pi and Pj and take these random variables mutually independent. If transports are possible or impossible independently with probability p and 1-p, respectively, then we give a lower bound for the distribution function of the smallest path at point log...

Total Progeny in Killed Branching Random Walk

We consider a branching random walk for which the maximum position of a particle in the n’th generation, Rn, has zero speed on the linear scale: Rn/n → 0 as n → ∞. We further remove (“kill”) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite [26, 31]. In this paper, we confirm a conjecture of Al...

The Martin Boundary for Random Walk

t-Martin boundary of killed random walks in the quadrant

We compute the t-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete t-harmonic functions. Our approach is uniform in t, and shows that there are three regimes for the Martin boundary.