An extension problem for the fractional derivative defined by Marchaud

TL;DR

This paper establishes a connection between the Marchaud fractional derivative and a parabolic extension problem, deriving properties and a Harnack principle for functions stationary under this derivative.

Contribution

It introduces a novel extension problem for the Marchaud fractional derivative, linking nonlocal and local operators and deriving new properties.

Findings

Derived a parabolic extension problem for Marchaud fractional derivative

Proved a Harnack principle for Marchaud-stationary functions

Connected properties of the fractional derivative to local operator properties

Abstract

We prove that the (nonlocal) Marchaud fractional derivative in $\mathbb{R}$ can be obtained from a parabolic extension problem with an extra (positive) variable, as the operator that maps the heat conduction equation to the Neumann condition. Some properties of the fractional derivative are deduced from those of the local operator. In particular we prove a Harnack principle for Marchaud-stationary functions.