Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics

We study boundary non-crossing probabilities $$\begin{aligned} P_{f,u} := \mathrm {P}\big (\forall t\in {\mathbb {T}}\ X_t + f(t)\le u(t)\big ) \end{aligned}$$ P f , u : = P ( ∀ t ∈ T X t + f ( t ) ≤ u ( t ) ) for a continuous centered Gaussian process X indexed by some arbitrary compact separable metric space $${\mathbb {T}}$$ T . We obtain both upper and lower bounds for $$P_{f,u}$$ P f , u . The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case $$P_{{y}f,u}$$ P y f , u , $${y}\rightarrow +\infty $$ y → + ∞ , which are two-term approximations (up to $$o({y})$$ o ( y ) ). The asymptotics are formulated in terms of the solution $${\tilde{f}}$$ f ~ to the constrained optimization problem $$\begin{aligned} \left\Vert h\right\Vert _{{\mathbb {H}}_X}\rightarrow \min , \quad h\in {\mathbb {H}}_X, h\ge f \end{aligned}$$ h H X → min , h ∈ H X , h ≥ f in the reproducing kernel Hilbert space $${\mathbb {H}}_X$$ H X of X. Several applications of the results are further presented.