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April 2003 Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion
Xia Chen
Ann. Probab. 31(2): 1052-1077 (April 2003). DOI: 10.1214/aop/1048516545


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.


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Xia Chen. "Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion." Ann. Probab. 31 (2) 1052 - 1077, April 2003.


Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1030.60026
MathSciNet: MR1964958
Digital Object Identifier: 10.1214/aop/1048516545

Primary: 60G15
Secondary: 60J25 , 60J60

Keywords: $m$-fold integrated Brownian motion , principal eigenvalues , quadratic functionals , small ball probabilities

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 2 • April 2003
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