First Passage Time for a Particular Gaussian Process

We find an explicit formula for the first passage probability, Qₐ(T | x) = Pᵣ(S(t) < a, 0 t T | S(0) = x), for all $T > 0$, where S is the Gaussian process with mean zero and covariance ES(τ)S(t) = max (1 - |t - τ|, 0). Previously, Qₐ(T x) was known only for T 1. In particular for $T = n$ an integer and -∞ < x < a < ∞, Qₐ(T x) = 1/φ(x) ∫D ⋯ ∫ φ(yᵢ - yⱼ₊₁ + a) dy₂ ⋯ dyₙ₊₁, where the integral is an n-fold integral on y₂, ⋯, yₙ₊₁ over the region D given by D = - x < y₂ < y₁ < ⋯ < yₙ₊₁\ and the determinant is of size (n + 1) × (n + 1), 0 < i, j n, with y₀ ≡ 0, y₁ ≡ a - x.