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. 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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Brownian motion Bessel process Hitting time Tauberian theorem excursions

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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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