Boundary Non-Crossing Probabilities Of Gaussian Processes: Sharp Bounds And Asymptotics

We study boundary non-crossing probabilities for a continuous centered Gaussian process X indexed by some arbitrary compact separable metric space . We obtain both upper and lower bounds for . The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case , , which are two-term approximations (up to ). The asymptotics are formulated in terms of the solution to the constrained optimization problem in the reproducing kernel Hilbert space of X. Several applications of the results are further presented.