Stretched non-local Pearson diffusions

Luisa Beghin

doi:10.1016/j.spa.2025.104854

Stretched non-local Pearson diffusions

2025-12-01

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DOI：10.1016/j.spa.2025.104854

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Abstract

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We define a novel class of time-changed Pearson diffusions, termed stretched non-local Pearson diffusions, where the stochastic time-change model has the Kilbas-Saigo function as its Laplace transform. Moreover, we introduce a stretched variant of the Caputo fractional derivative and prove that its eigenfunction is, in fact, the Kilbas-Saigo function. Furthermore, we solve fractional Cauchy problems involving the generator of the Pearson diffusion and the Fokker-Planck operator, providing both analytic and stochastic solutions, which connect the newly defined process and fractional operator with the Kilbas-Saigo function. We also prove that stretched non-local Pearson diffusions share the same limiting distributions as their standard counterparts. Finally, we investigate fractional hyperbolic Cauchy problems for Pearson diffusions, which resemble time-fractional telegraph equations, and provide both analytical and stochastic solutions. As a byproduct of our analysis, we derive a novel representation and an asymptotic formula for the Kilbas-Saigo function with complex arguments, which, to the best of our knowledge, are not currently available in the existing literature.

Keywords:

Pearson diffusion

Kilbas-Saigo function

Caputo fractional derivative

Fractional cauchy problem

Fractional diffusion

Stretched non-local Pearson diffusions

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