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For a recurrent linear diffusion on $\R_+$ we study the asymptotics of thedistribution 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 issubexponential this distribution behaves asymtotically as a multiple of theL\'evy measure. Using spectral representations we find the exact value of themultiple. For this we also need a result on the asymptotic behavior of theconvolution of a subexponential distribution and an arbitrary distribution on$\R_+.$ The exact knowledge of the asymptotic behavior of the distribution ofthe local time allows us to analyze the process derived via a penalizationprocedure with the local time. This result generalizes the penalizationsobtained in Roynette, Vallois and Yor \cite{rvyV} for Bessel processes.

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