The Annals of Probability

The Spectral Decomposition of a Diffusion Hitting Time

John T. Kent

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60E07: Infinitely divisible distributions; stable distributions 34B25

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


Kent, John T. The Spectral Decomposition of a Diffusion Hitting Time. Ann. Probab. 10 (1982), no. 1, 207--219. doi:10.1214/aop/1176993924.

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