Small Deviations for Gaussian Markov Processes Under the Sup-Norm

Abstract

Let {X(t); 0≤t≤1} be a real-valued continuous Gaussian Markov process with mean zero and covariance σ(s, t) = EX(s) X(t) ≠ 0 for 0 0, H>0 and G/H nondecreasing on the interval (0, 1). We show that $$\mathop {\lim }\limits_{\varepsilon \to 0} \varepsilon ^2 \log P({\text{ }}\mathop {\sup }\limits_{0 < t \leqslant 1} {\text{ |}}X(t)| < \varepsilon ) = - (\pi ^2 /8)\int_0^1 {(G'H - H'G)dt} $$ In the critical case, i.e. this integral is infinite, we provide the correct rate (up to a constant) for log P(sup0<t≤1 |X(t)|<∈) as ∈→0 under regularity conditions.