We prove results on exact asymptotics of the probabilities

where 2 ≤ p ≤ ∞, for two types of Gaussian processes η(t), namely, a stationary Ornstein-Uhlenbeck process and a Gaussian diffusion process satisfying the stochastic differential equation Derivation of the results is based on the principle of comparison with a Wiener process and Girsanov’s absolute continuity theorem. Original Russian Text ? V.R. Fatalov, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 2, pp. 75–95. Supported in part by the Russian Foundation for Basic Research, project no. 04-01-00700.