  Many important index transforms can be constructed via the spectral theory of
Sturm-Liouville differential operators. Using the spectral expansion method, we
investigate the general connection between the index transforms and the
associated parabolic partial differential equations. We show that the notion of
Yor integral, recently introduced by the second author, can be extended to the
class of Sturm-Liouville integral transforms.
  We furthermore show that, by means of the Feynman-Kac theorem, index
transforms can be used for studying Markovian diffusion processes. This gives
rise to new applications of index transforms to problems in mathematical
finance.
