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Spectral representation of transition density of Fisher–Snedecor diffusion

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 E[X^{;m};X^{;n};], 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.

Fisher–Snedecor polynomials; heavy-tailed diffusion; hypergeometric function; infinitesimal generator; Sturm–Liouville equation; transition density

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