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

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size $h>0$ $$(-\Delta_h)^su=f,$$ for $u,f:\mathbb{Z}_h\to\mathbb{R}$, $0<s<1$, is performed. The pointwise nonlocal formula for $(-\Delta_h)^su$ and the nonlocal discrete mean value property for discrete $s$-harmonic functions are obtained. We observe that a characterization of $(-\Delta_h)^s$ as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete H\"older spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincar\'e inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution) $$u=(-\Delta_h)^{-s}f,$$ which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy--Littlewood--Sobolev inequality for $(-\Delta_h)^{-s}$. As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as $h\to0$ in H\"older 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 $$(-\Delta)^sU=F,$$ in $\mathbb{R}$, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of~$h$.