Small Deviations for Gaussian Markov Processes Under the Sup-Norm

Let {X(t); 0t1} be a real-valued continuous Gaussian Markov process with mean zero and covariance (s, t) = EX(s) X(t) 0 for 0<s, t<1. It is known that we can write (s, t) = G(min(s, t)) H(max(s, t)) with G>0, H>0 and G/H nondecreasing on the interval (0, 1). We show thatIn the critical case, i.e. this integral is infinite, we provide the correct rate (up to a constant) for log P(sup_0<t1 |X(t)|<) as 0 under regularity conditions.