Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as h→0 in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of h under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian(−Δ)sU=F, in R, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of h.