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July 2005 On dynamical Gaussian random walks
Davar Khoshnevisan, David A. Levin, Pedro J. Méndez-Hernández
Ann. Probab. 33(4): 1452-1478 (July 2005). DOI: 10.1214/009117904000001044


Motivated by the recent work of Benjamini, Häggström, Peres and Steif [Ann. Probab. 34 (2003) 1–34] on dynamical random walks, we do the following: (i) Prove that, after a suitable normalization, the dynamical Gaussian walk converges weakly to the Ornstein–Uhlenbeck process in classical Wiener space; (ii) derive sharp tail-asymptotics for the probabilities of large deviations of the said dynamical walk; and (iii) characterize (by way of an integral test) the minimal envelope(s) for the growth-rate of the dynamical Gaussian walk. This development also implies the tail capacity-estimates of Mountford for large deviations in classical Wiener space.

The results of this paper give a partial affirmative answer to the problem, raised in Benjamini, Häggström, Peres and Steif [Ann. Probab. 34 (2003) 1–34, Question 4], of whether there are precise connections between the OU process in classical Wiener space and dynamical random walks.


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Davar Khoshnevisan. David A. Levin. Pedro J. Méndez-Hernández. "On dynamical Gaussian random walks." Ann. Probab. 33 (4) 1452 - 1478, July 2005.


Published: July 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1090.60066
MathSciNet: MR2150195
Digital Object Identifier: 10.1214/009117904000001044

Primary: 28C20 , 60F10 , 60J05 , 60J25

Keywords: Dynamical walks , large deviations , the Ornstein–Uhlenbeck process in Wiener space , upper functions

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 4 • July 2005
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