Small deviations for Gaussian Markov processes under the sup-norm

Let {X(t);0 less than or equal to t less than or equal to 1} be a real-valued continuous Gaussian Markov process with mean zero and covariance sigma(s, t) = EX(s) X(t) not equal 0 for 0 < s, t < 1. It is known that we can write sigma(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 that lim(epsilon-->0) epsilon(2) log P((0 < t less than or equal to 1) sup \X(t)\ < epsilon) = - (pi(2)/8) integral(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(sup(0 < t less than or equal to 1) \X(t)\ < epsilon) as epsilon --> 0 under regularity conditions.