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

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 $m=+\infty.$