## Prof. Dr. Marcel Schmidt

### Researcher, Project leader

Universität Leipzig

E-mail: marcel.schmidt(at)math.uni-leipzig.de

Telephone: +49 341 97 - 32116

Homepage: https://www.mathcs.uni-leipzig.de/math/a…

## Project

**19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces
**59**Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

## Publications within SPP2026

Beschreibung

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima’s ergodic theorem for the harmonic functions in the domain of the Lp generator. Secondly we prove analogues of Yau’s and Karp’s Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain growth criteria must be constant. As consequence we give an integral criterion for recurrence.

Publisher | Springer |

Book | Dirichlet Forms and Related Topics |

Volume | IWDFRT 2022 |

Pages | 201–221 |

Link to preprint version | |

Link to published version |

**Related project(s):****59**Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

The present book deals with the spectral geometry of infinite graphs. This topic involves the interplay of three different subjects: geometry, the spectral theory of Laplacians and the heat flow of the underlying graph. These three subjects are brought together under the unifying perspective of Dirichlet forms. The spectral geometry of manifolds is a well-established field of mathematics. On manifolds, the focus is on how Riemannian geometry, the spectral theory of the Laplace–Beltrami operator, Brownian motion and heat evolution interact. In the last twenty years large parts of this theory have been subsumed within the framework of strongly local Dirichlet forms. Indeed, this point of view has proven extremely fruitful.

Publisher | Springer |

Book | Grundlehren der mathematischen Wissenschaften |

Volume | 358 |

Pages | 668 |

Link to preprint version | |

Link to published version |

**Related project(s):****59**Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

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):****19**Boundaries, 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):****19**Boundaries, 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):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

Journal | to appear in Mathematische Zeitschrift |

Link to preprint version |

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces