Quadratic Functionals and Small Ball Probabilities for the m-fold Brownian Motion
Date
2002
Journal Title
Journal ISSN
Volume Title
Publisher
Department of Mathematical Sciences
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.
Description
Keywords
The m-fold Integrated Brownian Motion, quadratic functionals, small ball probabilities, principal eigenvalues