Diffusion determines the recurrent graph

Abstract

Abstract. We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown by counterexamples this result is optimal. Without the recurrence assumption, the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass.These investigations provide discrete counterparts to studies of diffusion on Euclidean domains and manifolds initiated by Arendt and continued by Arendt / Biegert / ter Elst and Arendt / ter Elst.A crucial step in our considerations shows that order isomorphisms are actually unitary maps (up to a scaling) in our context.

Intrinsic Metrics on Graphs: A Survey

“…The next lemma shows that the combinatorial graph distance is equivalent to an intrinsic metric if and only if the graphs has bounded geometry, i.e., Deg is bounded. This was already noted in [22,54].…”

“…An abstract way to see that σ is in general not intrinsic is discussed in [54,Section 1]. Namely, the set of 1-Lipschitz continuous functions Lip ρ with respect to an intrinsic metric ρ with Lipschitz constant one is included in A and Lip ρ is closed under taking suprema.…”

Disjointness-preserving operators and isospectral Laplacians

“…While motivated in large part by the above-mentioned observation about the form of the known counterexamples, the current note also follows in the tradition of earlier works of one of the present authors [3,4,6], which have also been extended recently in other directions [28,29]. In these works, as here, one investigates operators having an intertwining property such as (1.1) but where unitarity is replaced by some other property.…”

“…These results are extended to smooth manifolds in [6], while [3] deals with the corresponding question on the space of continuous functions vanishing at the boundary and at infinity. In [28], a similar analysis was given on weighted discrete graphs; this was extended recently in [29] to the much more general abstract setting of pairs of Dirichlet forms intertwined by order isomorphisms on L 2 -spaces under a wide variety of assumptions.…”

Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

“…In addition to the themes mentioned above, we add that the past decades have seen a number of independent breakthroughs on related topics covering analysis on infinite models. Of special note and relevance are the following [ALP99, FW03, Car73, FKW90, GHK + 15, KLW13,KLSW15,MP84,PS12,Pet12].…”