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On subexponentiality of the Lévy measure of the inverse local time; with applications to penalizations. (English) Zbl 1192.60089

Summary: 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 B. Roynette, P. Vallois and M. Yor [Stud. Sci. Math. Hung. 45, No. 1, 67–124 (2008; Zbl 1164.60307)] for Bessel processes.

MSC:

60J60 Diffusion processes
60J65 Brownian motion

Citations:

Zbl 1164.60307
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