Integrated Brownian motions and exact $L_2$-small balls

Abstract

We will introduce a class of m-times integrated Brownian motions. The exact asymptotic expansions for the $L_2$-small ball probabilities will be discussed for members of this class, of which the usual m-times integrated Brownian motion is an example. Another example will be what we call an Euler-integrated Brownian motion. We will also find very sharp estimates for the asymptotics of the eigenvalues of the covariance operator of integrated Brownian motions and will, therefore, obtain exact, not just logarithmic, asymptotics.