## The Annals of Probability

### The Spectral Decomposition of a Diffusion Hitting Time

John T. Kent

#### Abstract

All first hitting times for a one-dimensional diffusion belong to the Bondesson class of infinitely divisible distributions on $\lbrack 0, \infty\rbrack$. A distribution in this class can be conveniently represented in terms of its canonical measure. In this paper we establish a link between the canonical measure of a hitting time and the spectral measure of the differential generator of the diffusion. In particular, it is shown that the derivative of the canonical measure with respect to natural scale (as a function of the point being hit) equals the spectral measure of the differential generator on a restricted interval. The canonical measure is then calculated for several examples arising from the Bessel diffusion process, including the inverse of a gamma variate and the Hartman-Watson mixing distribution.

#### Article information

Source
Ann. Probab., Volume 10, Number 1 (1982), 207-219.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993924

Digital Object Identifier
doi:10.1214/aop/1176993924

Mathematical Reviews number (MathSciNet)
MR637387

Zentralblatt MATH identifier
0483.60075

JSTOR