Approximation of solutions of fractional diffusions in compact metric measure spaces

Abstract

In this note we prove that the solutions to diffusions associated with fractional powers of the Laplacian in compact metric measure spaces can be obtained as limits of the solutions to particular rescalings of some non-local diffusions with integrable kernels. The abstract approach considered here has several particular and interesting instances. As an illustration of our results, we present the case of dyadic metric measure spaces where the existence of solutions was already been proven in Actis and Aimar (Fract Calc Appl Anal 18(3):762–788, 2015).