Electronic Journal of Probability

On subexponentiality of the Lévy measure of the inverse local time; with applications to penalizations

Paavo Salminen and Pierre Vallois

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Abstract

Abstract. For a recurrent linear diffusion on the positive real axis we study the asymptotics of the distribution of its local time at 0 as the time parameter tends to infinity. Under the assumption that the Lévy measure of the inverse local time is subexponential this distribution behaves asymptotically as a multiple of the Lévy measure. Using spectral representations we find the exact value of the multiple. For this we also need a result on the asymptotic behavior of the convolution of a subexponential distribution and an arbitrary distribution on the positive real axis. The exact knowledge of the asymptotic behavior of the distribution of the local time allows us to analyze the process derived via a penalization procedure with the local time. This result generalizes the penalizations obtained by Roynette, Vallois and Yor in Studia Sci. Math. Hungar. 45(1), 2008 for Bessel processes.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 67, 1963-1991.

Dates
Accepted: 17 September 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819528

Digital Object Identifier
doi:10.1214/EJP.v14-686

Mathematical Reviews number (MathSciNet)
MR2540855

Zentralblatt MATH identifier
1192.60089

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J65: Brownian motion [See also 58J65] 60J30

Keywords
Brownian motion Bessel process Hitting time Tauberian theorem excursions

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Salminen, Paavo; Vallois, Pierre. On subexponentiality of the Lévy measure of the inverse local time; with applications to penalizations. Electron. J. Probab. 14 (2009), paper no. 67, 1963--1991. doi:10.1214/EJP.v14-686. https://projecteuclid.org/euclid.ejp/1464819528


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