  Motivated by the recent work of Benjamini, Haggstrom, Peres, and Steif (2003)
on dynamical random walks, we: Prove that, after a suitable normalization, the
dynamical Gaussian walk converges weakly to the Ornstein-Uhlenbeck process in
classical Wiener space; derive sharp tail-asymptotics for the probabilities of
large deviations of the said dynamical walk; and characterize (by way of an
integral test) the minimal envelop(es) for the growth-rate of the dynamical
Gaussian walk. This development also implies the tail capacity-estimates of
Mountford (1992) for large deviations in classical Wiener space. The results of
this paper give a partial affirmative answer to the problem, raised in
Benjamini et al (2003, Question 4), of whether there are precise connections
between the OU process in classical Wiener space and dynamical random walks.
