Exact asymptotics of large deviations of stationary Ornstein-Uhlenbeck processes for L p-functionals, p > 0

Abstract

We prove a general result on the exact asymptotics of the probability $$P\left\{ {\int\limits_0^1 {\left| {\eta _\gamma (t)} \right|^p dt > u^p } } \right\}$$ as u ? ?, where p > 0, for a stationary Ornstein-Uhlenbeck process ? ?(t), i.e., a Gaussian Markov process with zero mean and with the covariance function E??(t)??(s), t, s ? ?, ? > 0. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type. For p = 1 and p = 2, explicit formulas for the asymptotics are given.