Small Deviations for Gaussian Markov Processes Under the Sup-Norm

\. INTRODUCTION Let {X(t); 0 < t < 1} be a real-valued mean-zero Gaussian process, and let "||.||" be a semi-norm on the space of real functions on [0, 1]. The so called "small ball estimates" or "small deviation estimates" refer to the asymptotic behavior of This type of problem is quite delicate, and the asymptotic decay rate in (1.1), up to a constant, depends heavily on the process X(t) and the Let {X(t); 0 < t < 1} be a real-valued continuous Gaussian Markov process with mean zero and covariance s(s, t) = EX(s) X(t) = 0 for 0 0, H > 0 and G/H nondecreasing on the interval (0, 1). We show that In the critical case, i.e. this integral is infinite, we provide the correct rate (up to a constant) for log P(sup 0 < t < 1 |X(t)| < e) as e->0 under regularity conditions .