On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusions (Q2866535)

The authors provide a unified treatment of the spectral analysis of heavy-tailed Kolmogorov-Pearson diffusions: the reciprocal (inverse) gamma, the Fisher Snedecor, and the skew-Student. This includes a review of already known unified expressions for eigenvalues, eigenfunctions, and spectral cutoff. In each case, the spectrum consists of a finite set of simple eigenvalues and a purely absolutely continuous spectrum, in the first two cases of multiplicity one, in the third case of multiplicity two, discrete and continuous spectra non-overlapping. Explicit expressions for the spectral representations of the transition densities are given.NEWLINENEWLINE In a large appendix, some general theory of one-dimensional diffusions is reviewed, in particular, concerning diffusions the resolvent of which is the product of two special monotone solutions of the associated Sturm-Liouville equation. For completeness of the spectral theory of Kolmogorov-Pearson diffusions, an overview of the known results on Ornstein-Uhlenbeck, Cox-Ingersoll-Ross, and Jacobi diffusions is included. As is well-known, these have a purely discrete spectrum.