Preprint:
On Dynamical Gaussian Random Walks
D. Khoshnevisan, D. A. Levin, and P. J.
MéndezHernández
Abstract.
Motivated by the recent work of Benjamini, Häggström,
Peres, and Steif (2003)
on dynamical random walks, we:
 Prove that, after a
suitable normalization,
the dynamical Gaussian walk converges weakly to the OrnsteinUhlenbeck
process in classical Wiener space;
 derive sharp tailasymptotics
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 growthrate of the
dynamical Gaussian walk.
This development also
implies the tail capacityestimates 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.
Keywords.
Dynamical walks, the OrnsteinUhlenbeck process in Wiener
space, large deviations, upper functions.
AMS Classification (2000)
60J25, 60J05, 60Fxx, 28C20.
Support. The research of D. Kh.
was supported in part by a grant from
the National Science Foundation.
Pre/EPrints. This paper is available in
Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 841120090, U.S.A.
davar@math.utah.edu

David Asher Levin
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 841120090, U.S.A.
levin@math.utah.edu

Pedro J. MéndezHernández
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 841120090, U.S.A.
mendez@math.utah.edu

Last Update: July 25, 2003
© 2003  Davar Khoshnevisan, David Asher Levin, and
Pedro J. MéndezHernández