A note on a.s. finiteness of perpetual integral functionals of   diffusions
  In this note, with the help of the boundary classification of diffusions, we derive a criterion of the convergence of perpetual integral functionals of transient real-valued diffusions. In the particular case of transient Bessel processes, we note that this criterion agrees with the one obtained via Jeulin's convergence lemma. 
Paavo Salminen, Marc Yor (PMA)