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Stretched non-local Pearson diffusions

Luisa Beghin, Nikolai Leonenko, Ivan Papić, Jayme Vaz

TL;DR

This work generalizes fractional diffusion for Pearson diffusions by introducing stretched non-local dynamics governed by a Caputo-type operator with stretching parameter and a Kilbas–Saigo time-change. It shows that the Kilbas–Saigo function, as an eigenfunction of the stretched operator, naturally appears in both first- and second-order temporal equations, enabling analytic KS-based solutions for fractional Cauchy and hyperbolic-type problems, while retaining the Pearson spatial structure. The authors develop a comprehensive spectral framework, derive convergence and asymptotic results for KS with complex arguments, and provide both analytic and stochastic representations of the stretched processes, including a stochastic time-change interpretation via a KS-Laplace transform. They demonstrate that stretched non-local Pearson diffusions share the same limiting distributions as their classical counterparts and extend the analysis to fractional hyperbolic Pearson diffusion, with stochastic representations linking KS dynamics to subordinated-time processes. Overall, the paper broadens fractional diffusion modeling with KS-based time changes, offering flexible relaxation dynamics and new tools for non-local temporal modeling in diffusion-type systems.

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.

Stretched non-local Pearson diffusions

TL;DR

This work generalizes fractional diffusion for Pearson diffusions by introducing stretched non-local dynamics governed by a Caputo-type operator with stretching parameter and a Kilbas–Saigo time-change. It shows that the Kilbas–Saigo function, as an eigenfunction of the stretched operator, naturally appears in both first- and second-order temporal equations, enabling analytic KS-based solutions for fractional Cauchy and hyperbolic-type problems, while retaining the Pearson spatial structure. The authors develop a comprehensive spectral framework, derive convergence and asymptotic results for KS with complex arguments, and provide both analytic and stochastic representations of the stretched processes, including a stochastic time-change interpretation via a KS-Laplace transform. They demonstrate that stretched non-local Pearson diffusions share the same limiting distributions as their classical counterparts and extend the analysis to fractional hyperbolic Pearson diffusion, with stochastic representations linking KS dynamics to subordinated-time processes. Overall, the paper broadens fractional diffusion modeling with KS-based time changes, offering flexible relaxation dynamics and new tools for non-local temporal modeling in diffusion-type systems.

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.
Paper Structure (11 sections, 16 theorems, 216 equations, 2 figures)

This paper contains 11 sections, 16 theorems, 216 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Kilbas-Saigo function and the differential operator $\boldsymbol{\mathcal{D}^{(\alpha,\gamma)}_t}$
  3. A first-order equation involving $\boldsymbol{\mathcal{D}^{(\alpha,\gamma)}_t}$ and the KS function
  4. A second-order equation involving $\boldsymbol{\mathcal{D}^{(\alpha,\gamma)}_t}$ and the KS function
  5. Pearson diffusions and their classification
  6. Stretched non-local Pearson diffusions
  7. Fractional hyperbolic Pearson diffusion
  8. Stochastic representation
  9. Stochastic representation for stretched non-local Pearson diffusion
  10. Stochastic representation for fractional hyperbolic Pearson diffusion
  11. Double gamma function

Key Result

Theorem 2.2

Let $z \in \mathbb{C}$ be such that $\operatorname{Re}(z)>0$. Then the following formula is valid and integral on the right-hand side is convergent for any such $z$: with $\tau = 1/(am) > 0$.

Figures (2)

  • Figure 1: Integration contour for \ref{['M-B.KS.1']}.
  • Figure 2: Integration contour for \ref{['asymptotic.aux']}.

Theorems & Definitions (44)

  • Definition 1
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 34 more