On subexponentiality of the L\'evy measure of the diffusion inverse local time; with applications to penalizations

For a recurrent linear diffusion on $\R_+$ 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 L\'evy measure of the inverse local time is subexponential this distribution behaves asymtotically as a multiple of the L\'evy 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 $\R_+.$ 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 penalizations obtained in Roynette, Vallois and Yor \cite{rvyV} for Bessel processes.