# Members & Guests

## Dr. Marcel Schmidt

### Researcher

Friedrich-Schiller-Universität Jena

E-mail: schmidt.marcel(at)uni-jena.de
Telephone: +49 3641 9 46 113
Homepage: http://www.analysis-lenz.uni-jena.de/Tea…

## Project

19Boundaries, Greens formulae and harmonic functions for graphs 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

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.