Quadratic Functionals and Small Ball Probabilities for the m-fold Brownian Motion

Abstract

Let the Gaussian process Xm(t) be the m-fold integrated Brownian motion for positive integer m. The Laplace transform of the quadratic functional of Xm(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 Gaussian process. The connection is discovered by introducing an independent random shift. Various interplay between our results and principal eigenvalues for non-uniform elliptic generators on an unbounded domain are discussed.