On Maximizing the Speed of a Random Walk in Fixed Environments

We consider a random walk in a fixed Z environment composed of two point types: (q,1-q) and (p,1-p) for 1/2<q<p. We study the expected hitting time at N for a given number k of p-drifts in the interval [1,N-1], and find that this time is minimized asymptotically by equally spaced p-drifts.


Introduction
Procaccia and Rosenthal [1] studied how to optimally place given number of vertices with a positive drift on top of a simple random walk to minimize the expected crossing time of an interval. They ask about extending their work to the situation where the environment on Z is composed of two point types: (q, 1 − q) and (p, 1 − p) for 1 2 < q < p. This is the goal of this note. See [1] for background and further related work.
Consider nearest neighbor random walks on 0, 1, ..., N with reflection at the origin. We denote the random walk by {X n } ∞ n=0 , and by ω (i) the transition probability at vertex i: First, we prove the following proposition concerning the expected hitting time at vertex N : Proposition 1. For a walk ω starting at x, the hitting time T N = min {n ≥ 0|X n = N } satisfies: , and E x ω (T N ) stands for the expected hitting time. In particular: Corollary 2. The expected hitting time from 0 to N is symmetric under reflection of the environment, i.e. taking the environment ω Next we turn to the case of an environment consisting of two types of drifts, (q, 1 − q) (i.e. probability q to go to the right and 1 − q to the left) and (p, 1 − p), for some 1 2 < q < p ≤ 1. Assume that k of the vertices are p-drifts, and the rest are q-drifts. In [1] it was proven that for q = 1 2 equally spaced p-drifts minimize (for large N ). In this paper we extend this result for q > 1 2 . We define an environment in which the p-drifts are equally spaced (up to integer effects): and prove the following theorem: Theorem 3. For every ε > 0 there exists n 0 such that for every N > n 0 and environment ω: where k is the number of p-drifts in ω.
Finally, we consider the set of environments ω ak,k for some a ∈ N, and calculate Proposition 4. Let a ∈ N. Then:

Proof of the main theorem
Proof of Proposition 1 . Define By conditioning on the first step: To solve these equations, define Next define ρ k for k in the circle Z N −1 , such that for 1 ≤ k ≤ N − 1 we will have ρ k = ρ k (gluing the point 0 to the point N − 1), and then look at: This way, rather than summing Proposition 6. Define α = 1−q q , β = 1−p p . Since β < α < 1: for some constant C (α) which doesn't depend on N . Since every drift appears in d intervals of length d, Proof. For convenience, we omit d from the notation, and set n = (n 1 , ..., n N −1 ). If a vector n satisfies n i − n j ≤ 1 ∀i, j, we say n is almost constant. We will show that σ is minimal for some almost constant vector. Then we show that σ takes on the same value for all almost constant vectors under the restriction, and this completes the proof. Suppose σ is minimized (under the restriction) by some vector n 0 . If n 0 is almost constant, we are done. Else, for some i, j we have that n 0 i − n 0 j ≥ 2. We choose i, j such that n 0 i − n 0 j is maximal. Define: n 1 satisfies the restriction, and σ n 0 ≥ σ n 1 : where the inequality follows from the fact that 0 ≤ β α < 1 and n 0 j < n 0 i − 1. From minimality of σ n 0 , we get that σ n 1 is also minimal. This process must end after a finite number of steps f , yielding an almost constant n f which minimizes σ.
Now for a general almost constant vector n, set a = min {n l : 1 ≤ l ≤ N − 1}. We have n l ∈ {a, a + 1}, so defining m 0 to be the number of a's and m 1 to be the number of a + 1's, we get: and since m 1 < N − 1, there is a unique solution for natural a, m 1 . So all almost constant n (satisfying the restriction) are the same up to ordering, and since σ doesn't depend on the order, they all give the same value.
Claim 9. For every choice of M, k, the placement of k drifts on the circle Z M in which the ith drift is at the point i · M σ d , and by claims 8 and 9 each σ d is minimized by this configuration, therefore the sum is also minimized.
Proof of Theorem 3. From Proposition 6, 0 < S N − S N < C. Let n 0 = 2C ε . Then for N > n 0 : where we denote by S * N and S * N the values caculated for ω N,k .

Further questions
(1) Show that the optimal environment also minimizes the variance of the hitting time.
(2) Can this result be extended to a random walk on Z with a given density of drifts (as in [1])? (3) Can similar results be found for other graphs? For example, Z 2 × Z N .