Subordination principle, Wright functions and large-time behaviour for the discrete in time fractional diffusion equation

TL;DR

This paper investigates the large-time asymptotic behavior of solutions to a discrete in time fractional diffusion equation using subordination formulas involving Wright functions and Fox H-functions.

Contribution

It introduces new subordination formulas with discrete Gaussian kernels and Wright functions to analyze fractional discrete diffusion equations.

Findings

Asymptotic behavior characterized in $L^p$ spaces.

New subordination formulas derived for discrete fractional diffusion.

Representation via Fox H-functions established.

Abstract

The main goal in this paper is to study asymptotic behaviour in $L^p(\mathbb{R}^N)$ for the solutions of the fractional version of the discrete in time $N$-dimensional diffusion equation, which involves the Caputo fractional $h$-difference operator. The techniques to prove the results are based in new subordination formulas involving the discrete in time Gaussian kernel, and which are defined via an analogue in discrete time setting of the scaled Wright functions. Moreover, we get an equivalent representation of that subordination formula by Fox H-functions.