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

Paavo P. Salminen (1) , Pierre P. Vallois (2)

1 MD - Mathematical department [Abo Finland]
2 IECN - Institut Élie Cartan de Nancy

For a recurrent linear diffusion on $\Bbb R_+$ 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 representation 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 $\Bbb R_+$. 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 penalization procedure with the local time. This result generalizes the penalizations obtained in Roynette, Vallois and Yor [22] for Bessel processes.

Mots clés: Brownian motion; Bessel process; Hitting time; Tauberian theorem; excursions

Domaines: Probabilités [math.PR]

Paavo P. Salminen, Pierre P. Vallois. On subexponentiality of the Lévy measure of the diffusion inverse local time; with applications to penalizations. Electronic Journal of Probability, 2009, 14 (67), pp.1963-1991. ⟨hal-00285718⟩