Spectral representation of transition density of Fisher-Snedecor diffusion
Résumé
We analyse spectral properties of an ergodic heavy-tailed diffusion with the Fisher-Snedecor invariant distribution and compute spectral representation of its transition density. The spectral representation is given in terms of a sum involving finitely many eigenvalues and eigenfunctions (Fisher-Snedecor orthogonal polynomials) and an integral over the absolutely continuous spectrum of the corresponding Sturm-Liouville operator. This result enables the computation of the two-dimensional density of the Fisher-Snedecor diffusion as well as calculation of moments of the form, where m and n are at most equal to the number of Fisher-Snedecor polynomials. This result is particularly important for explicit calculations associated with this process. © 2013 Copyright Taylor and Francis Group, LLC.