On (Global) Unique Continuation Properties of the Fractional Discrete Laplacian

Abstract: We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique continuation properties, the fractional discrete Laplacian does not enjoy these in general. While discretization thus counteracts the strong rigidity properties of the continuum fractional Laplacian, by discussing quantitative forms of unique continuation, we illustrate that these properties can be recovered if exponentially small (in the lattice size) correction terms are added. This in particular allows us to deduce uniform stability properties for a discrete, linear inverse problem for the fractional Laplacian. We complement these observations with a transference principle and the discussion of these properties on the discrete torus.