Open Access
Translator Disclaimer
February, 1982 The Spectral Decomposition of a Diffusion Hitting Time
John T. Kent
Ann. Probab. 10(1): 207-219 (February, 1982). DOI: 10.1214/aop/1176993924


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.


Download Citation

John T. Kent. "The Spectral Decomposition of a Diffusion Hitting Time." Ann. Probab. 10 (1) 207 - 219, February, 1982.


Published: February, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0483.60075
MathSciNet: MR637387
Digital Object Identifier: 10.1214/aop/1176993924

Primary: 60J60
Secondary: 34B25 , 60E07

Keywords: $t$-distribution , canonical measure , Diffusion hitting time , generalized convolutions of mixtures of exponential distributions , Hartman-Watson mixing distribution , Infinite divisibility , spectral measure , von Mises-Fisher distribution

Rights: Copyright © 1982 Institute of Mathematical Statistics


Vol.10 • No. 1 • February, 1982
Back to Top