Some results on regularity and monotonicity of the speed for excited random walks 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) in dimension d≥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≤d≠3. However, this is not enough to imply that the speed is increasing in a neighborhood of 0. It still remains to prove that the derivative is continuous at 0. Moreover, this paper gives some results of monotonicity for m-excited random walk when m is large enough.