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.
Publié le : 2003-04-14
Classification:
$m$-fold integrated Brownian motion,
quadratic functionals,
small ball probabilities,
principal eigenvalues,
60G15,
60J25,
60J60
@article{1048516545,
author = {Chen, Xia},
title = {Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion},
journal = {Ann. Probab.},
volume = {31},
number = {1},
year = {2003},
pages = { 1052-1077},
language = {en},
url = {http://dml.mathdoc.fr/item/1048516545}
}
Chen, Xia. Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion. Ann. Probab., Tome 31 (2003) no. 1, pp. 1052-1077. http://gdmltest.u-ga.fr/item/1048516545/