Stretched non-local Pearson diffusions

Beghin, Luisa, Leonenko, Nikolai ORCID: https://orcid.org/0000-0003-1932-4091, Papaic, Ivan and Vaz, Jayme 2026. Stretched non-local Pearson diffusions. Stochastic Processes and their Applications 195 , 104854. 10.1016/j.spa.2025.104854

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Abstract

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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Schools > Mathematics
Additional Information:	RRS applied 07/01/2026 AB
Publisher:	Elsevier
ISSN:	0304-4149
Date of First Compliant Deposit:	16 December 2025
Date of Acceptance:	15 December 2025
Last Modified:	07 Jan 2026 14:30
URI:	https://orca.cardiff.ac.uk/id/eprint/183288

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