Non-Local Pearson diffusions

TL;DR

This paper investigates non-local heat equations involving Bernstein functions and Pearson diffusion generators, providing explicit solutions, stochastic representations, and analyzing properties like limit distributions and dependence structures.

Contribution

It introduces explicit solutions and stochastic representations for non-local Pearson diffusion equations, expanding understanding of their probabilistic and analytical properties.

Findings

Explicit solutions via spectral decomposition.

Stochastic representations using time-changed Pearson diffusions.

Analysis of limit distributions and dependence properties.

Abstract

In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker-Planck operator of a Pearson diffusion. Such kind of non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusion to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence.