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Ultracontractive Properties for Directed Graph Semigroups with Applications to Coupled Oscillators

Published online by Cambridge University Press:  31 October 2019

Jason J. Bramburger
Jason J. Bramburger*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, R.I. Email: jason_bramburger@brown.edu
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Abstract

It is now well known that ultracontractive properties of semigroups with infinitesimal generator given by an undirected graph Laplacian operator can be obtained through an understanding of the geometry of the underlying infinite weighted graph. The aim of this work is to extend these results to semigroups with infinitesimal generators given by a directed graph Laplacian operator through an analogous inspection of the geometry of the underlying directed graph. In particular, we introduce appropriate nomenclature to discuss the geometry of an infinite directed graph, as well as provide sufficient conditions to extend ultracontractive properties of undirected graph Laplacians to those of the directed variety. Such directed graph Laplacians can often be observed in the study of coupled oscillators, where recent work made explicit the link between synchronous patterns to systems of identically coupled oscillators and ultracontractive properties of undirected graph semigroups. Therefore, in this work we demonstrate the applicability of our results on directed graph semigroups by extending the aforementioned investigation beyond the idealized case of identically coupled oscillators.

Keywords

Directed graphheat kernelcoupled oscillatorsgraph Laplacian
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 1 , March 2020 , pp. 13 - 30
Copyright
© Canadian Mathematical Society 2019

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Footnotes

This work was supported by an NSERC PDF.

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