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Many important index transforms can be constructed via the spectral theory ofSturm-Liouville differential operators. Using the spectral expansion method, weinvestigate the general connection between the index transforms and theassociated parabolic partial differential equations. We show that the notion ofYor integral, recently introduced by the second author, can be extended to theclass of Sturm-Liouville integral transforms.  We furthermore show that, by means of the Feynman-Kac theorem, indextransforms can be used for studying Markovian diffusion processes. This givesrise to new applications of index transforms to problems in mathematicalfinance.

The spectral expansion approach to index transforms and connections with the theory of diffusion processes