Excursions of a Gaussian process with variable variance above a barrier increasing to infinity

For a family of real-valued Gaussian processes ξ _u (t), t ∈ 0, T], we obtain an exact asymptotics of the probability of crossing a level u as u → ∞ under certain conditions on the variance and correlation. This result is applied to the investigation of excursions of a stationary zero-mean process above a barrier increasing to infinity.