Prof. Dr. Daniel Lenz
Project leader

Friedrich-Schiller-Universität Jena
E-mail: daniel.lenz(at)uni-jena.de
Telephone: +49 3641 9 46 131
Homepage: http://www.analysis-lenz.uni-jena.de/
Project
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces
Publications within SPP2026
We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods.
Journal | Journal de Mathématiques Pures et Appliquées. (9) |
Volume | 126 |
Pages | 109--143 |
Link to preprint version | |
Link to published version |
Related project(s):
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces
We study pairs of Dirichlet forms related by an intertwining order
isomorphisms between the associated \(L^2\)-spaces. We consider the
measurable, the topological and the geometric setting respectively.
In the measurable setting, we deal with arbitrary (irreducible)
Dirichlet forms and show that any intertwining order isomorphism is
necessarily unitary (up to a constant). In the topological setting
we deal with quasi-regular forms and show that any intertwining
order isomorphism induces a quasi-homeomorphism between the
underlying spaces. In the geometric setting we deal with both
regular Dirichlet forms as well as resistance forms and essentially
show that the geometry defined by these forms is preserved by
intertwining order isomorphisms. In particular, we prove in the
strongly local regular case that intertwining order isomorphisms
induce isometries with respect to the intrinsic metrics between the
underlying spaces under fairly mild assumptions. This applies to a
wide variety of metric measure spaces including
\(\mathrm{RCD}(K,N)\)-spaces, complete weighted Riemannian manifolds
and complete quantum graphs. In the non-local regular case our
results cover in particular graphs as well as fractional Laplacians
as arising in the treatment of \(\alpha\)-stable Lévy processes. For
resistance forms we show that intertwining order isomorphisms are
isometries with respect to the resistance metrics.
Our results can can be understood as saying that diffusion always
determines the Hilbert space, and -- under natural compatibility
assumptions -- the topology and the geometry respectively. As special
instances they cover earlier results for manifolds and graphs.
Related project(s):
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces
We study topological Poincaré type inequalities on generalgraphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory ofLaplacians once a probability measure on the graph is chosen. More specifically,we obtain a variational characterization of these constants as infimum over spectral gaps of all Laplacians on the graphs associated to probability measures.
Related project(s):
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces
Journal | to appear in Mathematische Zeitschrift |
Link to preprint version |
Related project(s):
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces