## The Annals of Probability

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

Xia Chen

#### Abstract

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

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

Dates
First available in Project Euclid: 24 March 2003

https://projecteuclid.org/euclid.aop/1048516545

Digital Object Identifier
doi:10.1214/aop/1048516545

Mathematical Reviews number (MathSciNet)
MR1964958

Zentralblatt MATH identifier
1030.60026

#### Citation

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

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• KNOXVILLE, TENNESSEE 37996 E-MAIL: xchen@math.utk.edu DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF DELAWARE
• NEWARK, DELAWARE 19716 E-MAIL: wli@math.udel.edu