Stretched non-local Pearson diffusions

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 argument, which, to the best of our knowledge, are not currently available in the existing literature.