Small ball probabilities for Gaussian Markov processes under the Lp-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<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 L p -norm on C[0,1], 1⩽ p⩽∞ lim ε→0 ε 2 log P(||X(t)|| p<ε)=−κ p ∫ 0 1 (G′H−H′G) p/(2+p) dt (2+p)/p and its various extensions.