The Annals of Probability
- Ann. Probab.
- Volume 31, Number 2 (2003), 1052-1077.
Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion
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.
Ann. Probab., Volume 31, Number 2 (2003), 1052-1077.
First available in Project Euclid: 24 March 2003
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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. https://projecteuclid.org/euclid.aop/1048516545