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

Cong-Dan Pham

Stochastic Processes and their Applications, 2019, vol. 129, issue 7, 2286-2319

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.

Date: 2019
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DOI: 10.1016/j.spa.2018.06.015

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