ISSN 0032-9460, Problems of Information Transmission, 2014, Vol. 50, No. 4, pp. 371–389.
Pleiades Publishing, Inc., 2014.
Original Russian Text
V.R. Fatalov, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 4, pp. 79–99.
Gaussian Ornstein–Uhlenbeck and Bogoliubov
Processes: Asymptotics of Small Deviations
-Functionals, 0 <p<∞
V. R. Fatalov
Laboratory of Probability, Faculty of Mechanics and Mathematics,
Lomonosov Moscow State University, Moscow, Russia
Received September 17, 2014
Abstract—We prove results on sharp asymptotics of probabilities
dt < ε
where 0 <p<∞, for three Gaussian processes X(t), namely the stationary and nonstationary
Ornstein–Uhlenbeck process and the Bogoliubov process. The analysis is based on the Laplace
method for sojourn times of a Wiener process.
1. INTRODUCTION AND FORMULATION OF THE MAIN RESULTS
Ornstein–Uhlenbeck processes are known to be a background for many models used in various
problems of theory of stochastic processes, theoretical physics, statistical radio engineering, and
a number of other ﬁelds of natural science (see, e.g., [1–11]). The Gaussian Bogoliubov process,
which plays an important role in equilibrium statistical mechanics (see ), is deﬁned and partially
analyzed in . Though these two classes of processes are closely related to a Wiener process,
ﬁnding exact distributions of various nonlinear functionals of these processes is nevertheless a very
complicated problem. In particular, of great theoretical and practical interest is the problem of
approximate evaluation of distributions of such functionals as
dt,0<p<∞ (which we
refer to as the L
|x(t)|, etc. In [14–19], based on various modiﬁcations of the
Laplace method in functional spaces, approximate (asymptotic) formulas for these distributions
are obtained in the regions of large and small values (see also ).
In the present paper we prove new results on sharp small deviation asymptotics of Ornstein–
Uhlenbeck and Bogoliubov process for L
-functional, 0 <p<∞.Thecaseof2≤ p<∞ was
previously studied in [15, 17, 19]. The results are obtained with the use of a new general theorem
on asymptotics of one model Laplace-type integral for sojourn times of a Wiener process.
Now we pass to precise formulations of the results. Throughout what follows, [Ω,F, P]isarich
enough basic probability space, and E is expectation with respect to the probability measure P.
Supported in part by the Russian Foundation for Basic Research, project no. 11-01-00050.