The Annals of Probability

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

Xia Chen

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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.

Article information

Ann. Probab., Volume 31, Number 2 (2003), 1052-1077.

First available in Project Euclid: 24 March 2003

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Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J60: Diffusion processes [See also 58J65]

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


Chen, Xia. Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion. Ann. Probab. 31 (2003), no. 2, 1052--1077. doi:10.1214/aop/1048516545.

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