CNS.23

Objective:

This project will investigate discrete counterparts for various types of qualitative and quantitative uniqueness properties, as well as functional inequalities, for solutions of equations involving a discrete Laplacian, fractional powers of a discrete Laplacian, or more generally Laplacians on graphs. We will test the degree of rigidness of certain qualitative uniqueness properties of solutions to continuous equations and the quantification of the failure of analogous properties on the lattice. The techniques employed will take as starting point their continuous counterparts, but we will also exploit spectral theory of Laplacians, which is understood within the framework of graphs or, more generally, in manifolds. We will exploit the fact that a discrete setting is a metric space supporting the counting measure to go forward and deal with Laplacians on a path graph equipped with a doubling measure. We will also study functional inequalities related to Laplacians associated to graphs. Particular models of graphs, like the so called spider graphs or Sierpinski triangle graphs will be studied in a more concrete way. As natural continuations, the proposal will open the possibility to explore the relation with birth-death processes and random walks on graphs, the analysis in graphs with different measures and the subsequent harmonic analysis associated with the relevant metric spaces.