Some results on regularity and monotonicity of the speed for excited random walk in low dimensions

TL;DR

This paper investigates the regularity and monotonicity of the speed of excited random walks in low dimensions, proving differentiability with respect to bias and exploring conditions for monotonicity.

Contribution

It establishes the differentiability of the speed in bias parameter for excited random walks in dimensions two and higher, including at the critical point zero.

Findings

Speed is infinitely differentiable in bias for d≥2.

Speed is differentiable and positive derivative at zero for d≠3.

Monotonicity results for large m-excited random walks.

Abstract

Using renewal times and Girsanov's transform, we prove that the speed of the excited random walk is infinitely differentiable with respect to the bias parameter in $(0,1)$ for the dimension $d\ge 2$. At the critical point $0$, using a special method, we also prove that the speed is differentiable and the derivative is positive for every dimension $2\leq d\neq 3.$ However, this is not enough to imply that the speed is increasing in a neighborhood of $0.$ It still remains to prove the derivative is continuous at $0$. Moreover, this paper gives some results of monotonicity for $m-$excited random walk when $m$ is large enough or $m=+\infty.$