Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion

Abstract

Let the Gaussian process $X_m(t)$ be the $m$-fold integrated Brownian motion for positive integer $m$. The Laplace transform of the quadratic functional of $X_m(t)$ is found by using an appropriate self-adjoint integral operator. The result is then used to show the power of a general connection between small ball probabilities for the Gaussian process. The connection is discovered by introducing an independent random shift. The interplay between our results and the principal eigenvalues for nonuniform elliptic generators on an unbounded domain is also discussed.