One-dimensional random walks, decreasing rearrangements and discrete Steiner symmetrization

Abstract

Take a simple random walk in the “blind alley” {1, 2, …, N + 1}, starting at 1, with the boundary condition that movement to the left of 1 results in staying put at 1. Each time the random walk visits a point n ∈ {1, 2, …, N}, it is subject to a danger and has a probability d n of being consumed by it. We prove that the probability of safe arrival at N + 1 is increased if the d n are replaced by their non-decreasing rearrangement d n #. Next, we consider the same random walk but now on all of Z +, again with a danger d n at each point n ∈ Z +. Let T d be the time of first absorption by one of the dangers d n . We prove that P( T d ≥ λ) ≤ P( T d# ≥ λ) for all λ ∈ Z +. Finally, we obtain a theorem on Steiner rearrangement and generalized discrete harmonic measure for discrete cases which are a priori symmetric under a reflection in an appropriate axis. Our methods are completely elementary.