How to approximate the fractional Laplacian by the fractional discrete Laplacian

We use the solution to the semi-discrete heat equation in combination with the methodology of semigroups to define and obtain the pointwise formula for the fractional powers of the discrete Laplacian in a mesh of size $h>0$. This operator corresponds to the process of a particle that is allowed to randomly jump to \textit{any} point in the mesh with a certain probability. It is shown that solutions to the continuous fractional Poisson equation $(-\Delta)^sU=F$ can be approximated by solutions to the fractional discrete Dirichlet problem $(-\Delta_h)^su=f$ in $B_R$, $u=0$ in $B_R^c$. We obtain novel error estimates in the strongest possible norm, namely, the $L^\infty$ norm, under minimal natural H\”older regularity assumptions. Key ingredients for the analysis are the regularity estimates for the fractional discrete Laplacian, which are of independent interest.
 
Biography: Pablo Raul Stinga obtained his degree in Mathematics (Licenciatura en Ciencias Matemáticas) with Honors at Universidad Nacional de San Luis, Argentina, in 2005. He attended graduate school at Universidad Autónoma de Madrid (UAM), in Spain. He received his Master’s in Mathematics and Applications in 2007. Stinga earned his Ph.D. in Mathematics with European Mention Summa Cum Laude at UAM under the direction of Jose L. Torrea in 2010. He then became a researcher at Universidad de Zaragoza (2010) and at Universidad de La Rioja (2011-2012), both in Spain. In 2012, he was appointed R.H. Bing Fellow Instructor No.1 at The University of Texas at Austin, where he worked in collaboration with Luis A. Caffarelli. In 2015, he became an Assistant Professor of Mathematics at Iowa State University (ISU). 
 
Stinga works in the broad area of analysis, with main focus in harmonic analysis and partial differential equations. More precisely, the analysis of fractional and nonlocal linear and nonlinear elliptic and parabolic PDEs, including discrete nonlocal equations, free boundary problems, and the harmonic analysis of orthogonal expansions, singular integral operators, related functional spaces and geometry of Banach spaces.