Abstract

We consider a nearest neighbour random walk on the integers with absorption at 0 and constant jump probabilities to the left and right of zero. The associated spectral radius is rho and the spatial rho-Martin exit boundary comprises two extremal points and associated minimal excessive functions h(-infinity) and h(+infinity) associated with sequences x(n) converging to -infinity and +infinity respectively. We construct a space-time sequence (x(n), n) with x(n) -> -infinity converging to a point in the space-time rho-Martin exit boundary whose associated space time harmonic function h(x, t) is minimal and of the form rho-t h(+infinity) +1 (x) not p-t(h-infinity)(x) as might have been hoped.