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

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  • Pham, Cong-Dan

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.

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  • Pham, Cong-Dan, 2019. "Some results on regularity and monotonicity of the speed for excited random walks in low dimensions," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2286-2319.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:7:p:2286-2319
    DOI: 10.1016/j.spa.2018.06.015
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    References listed on IDEAS

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    1. Lebowitz, Joel L. & Rost, Hermann, 1994. "The Einstein relation for the displacement of a test particle in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 54(2), pages 183-196, December.
    2. Holmes, Mark & Sun, Rongfeng, 2012. "A monotonicity property for random walk in a partially random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1369-1396.
    3. Komorowski, T. & Olla, S., 2005. "Einstein relation for random walks in random environments," Stochastic Processes and their Applications, Elsevier, vol. 115(8), pages 1279-1301, August.
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