Small ball probabilities for Gaussian Markov processes under the Lp-norm

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<s, t<1. It is known that we can write σ(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 for the Lp-norm on C[0,1], 1⩽p⩽∞limε→0ε2 log P(||X(t)||p<ε)=−κp∫01(G′H−H′G)p/(2+p) dt(2+p)/pand its various extensions.