On dynamical Gaussian random walks

Abstract

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.