Project Details
Projekt
Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces
Subject Area
Mathematics
Term
from 2017 to 2023
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 339133485
A locally compact separable metric space together with a regular Dirichlet form is called a Dirichlet space. Thus, a Dirichlet space is a metric measure space with an additional structure.Examples for Dirichlet spaces include (weighted) Riemannian manifolds, fractals and graphs. Any Dirichlet form comes with a selfadjoint operator, called the generator, and an associated Markov process. In the case of Riemannian manifolds the generator is the Laplace-Beltrami operator and the Markov process in question is Brownian motion. Similarly, in the case of fractals the generator is a Laplace type operator and the Markov process is a Brownian motion. In the case of graphs, the generator is the graph Laplacian and the associated Markov process is a jump process. In this way, Dirichlet forms provide an analytic description of (versions of) Brownian motion on the underlying topological space.In the setting of Dirichlet spaces there is a strong interplay between geometric properties of the space, spectral features of the generator of the Dirichlet form and stochastic features of the associated Markov process.In this project we study this interplay focusing on global properties viz. on properties of the geometry ``far out'' and corresponding spectral and stochastic features.Two approaches will be pursued. One approach is centered around the compactification via the Royden boundary, corresponding boundary terms and Greens formulae. Here, we aim ata) understanding the boundary as a metric boundary,b) describing the selfadjoint Markov extensions for general Dirichlet forms via Dirichlet forms on the boundary,c) describing selfadjoint extensions of the Laplacian on (bundles over) graphs via boundary conditions,d) characterize global properties of the underlying Markov process such as recurrence, stochastic completeness and Markov uniqueness by vanishing boundary terms.In the second approach features of the geometry “far out” are captured via generalized eigenfunctions. With this approach we intend toa) show absence of non-trivial harmonic functions with $L^{p}$ growth conditions on general Dirichlet spaces,b) bound the dimension of the space of polynomially bounded harmonic functions for graphs,c) understand the support of eigenfunctions,d) describe the decay properties of generalized eigenfunctions for graphs.The two approaches are connected and making this explicit is an additional part of the project.The project will focus on the non-smooth non-local situation of graphs. However, we will strive for arguments dealing with general Dirichlet spaces. So, the investigations on graphs can be seen as a first step towards the theory of general Dirichlet spaces.
DFG Programme
Priority Programmes
Subproject of
SPP 2026:
Geometry at Infinity
