Boundaries, Green's formulae and harmonic functions for graphs and Dirichlet spaces - follow up

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A locally compact separable metric space together with a regular Dirichlet form is called a Dirichlet space. Examples for Dirichlet spaces include (weighted) Riemannian manifolds, fractals and graphs. Any Dirichlet form comes with a self-adjoint 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 the Brownian motion. 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 by two approaches. One approach, focusing on global properties of the geometry and the corresponding spectral and stochastic features, is centered around compactification via the Royden boundary, boundary terms and Greens formulae.The other approach, focusing on more local features of the geometry, is centered around harmonic functions and (generalized) eigenfunctions. These approaches are strongly related and exploring their relationship will lead to additional insights. The project will focus on the non-smooth, non-local situation of graphs. However, we will strive for arguments dealing with general Dirichlet spaces.