An extension problem for the fractional derivative defined by Marchaud

We prove that the (nonlocal) Marchaud fractional derivative in 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 inequality for Marchaud-stationary functions.