Ultracontractive Properties for Directed Graph Semigroups with Applications to Coupled Oscillators

Bramburger, Jason J.

Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 13-30

Résumé

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.

Détail

DOI : 10.4153/S0008439519000390

Mots-clés : Directed graph, heat kernel, coupled oscillators, graph Laplacian

Bibliographie

[1] Barlow, M., Coulhon, T., and Grigor’Yan, A., Manifolds and graphs with slow heat kernel decay. Invent. Math. 144(2001), 609–649. https://doi.org/10.1007/s002220100139 Google Scholar

[2] Bramburger, J., Stability of infinite systems of coupled oscillators via random walks on weighted graphs. Trans. Amer. Math. Soc. 372(2019), 1159–1192. https://doi.org/10.1090/tran/7609 Google Scholar

[3] Chern, I.-L. and Liu, T.-P., Convergence to diffusion waves of solutions for viscous conservation laws. Comm. Math. Phys. 110(1987), 503–517. Google Scholar

[4] Chen, Y., Khong, S. Z., and Georgiou, T. T., On the definiteness of graph Laplacians with negative weights: Geometrical and passivity-based approaches. American Control Conference (ACC), 2016, pp. 2488–2493. Google Scholar

[5] Corinto, F., Bonnin, M., and Gilli, M., Weakly connected oscillatory network models for associative and dynamic memories. Internat J. Bifur. Chaos Appl. Sci. Engrg. 17(2007), 4365–4379. https://doi.org/10.1142/S0218127407020014 Google Scholar

[6] Coulhon, T. and Saloff-Coste, L., Isopérimétrie sure les groupes et les variétés. Rev. Mat. Iberoam. 9(1993), 293–314. https://doi.org/10.4171/RMI/138 Google Scholar

[7] Delmotte, T., Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoam. 15(1999), 181–232. https://doi.org/10.4171/RMI/254 Google Scholar

[8] Ermentrout, G. B., Stable periodic solutions to discrete and continuum arrays of weakly coupled nonlinear oscillators. SIAM J. Appl. Math. 52(1992), 1665–1687. https://doi.org/10.1137/0152096 Google Scholar

[9] Ermentrout, G. B. and Kopell, N., Multiple pulse interactions and averaging in systems of coupled neural oscillators. J. Math. Biol. 29(1991), 195–217. https://doi.org/10.1007/BF00160535 Google Scholar

[10] Folz, M., Gaussian upper bounds for heat kernels of continuous time simple random walks. Electron. J. Probab. 16(2011), 1693–1722. Google Scholar

[11] Frank, R. and Kovarik, H., Heat kernels of metric trees and applications. SIAM J. Math. Anal. 45(2013), 1027–1046. https://doi.org/10.1137/120886297 Google Scholar

[12] Grigor’Yan, A., Heat kernels on manifolds, graphs and fractals. In: European Congress of Mathematics, Vol. I (Barcelona, 2000). Prog. Math., 201, Birkäuser, Basel, 2001, pp. 393–406. Google Scholar

[13] Grigor’Yan, A. and Telcs, A., Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109(2001), 451–510. https://doi.org/10.1215/S0012-7094-01-10932-0 Google Scholar

[14] Gyrya, P. and Saloff-Coste, L., Neumann and Dirichlet heat kernels in inner uniform domains. Astérisque 336(2011), viii+144 pp. Google Scholar

[15] Hoffman, A., Hupkes, H. J., and Van Vleck, E. S., Multi-dimensional stability of waves travelling through rectangular lattices in rational directions. Trans. Amer. Math. Soc. 367(2015), 8757–8808. https://doi.org/10.1090/S0002-9947-2015-06392-2 Google Scholar

[16] Hoppensteadt, F. C. and Izhikevich, E. M., Weakly connected neural networks. Applied Mathematical Sciences, Vol. 126, Springer-Verlag, New York, 1997. https://doi.org/https://doi.org/10.1007/978-1-4612-1828-9 Google Scholar

[17] Horn, P., Lin, Y., Liu, S., and Yau, S.-T., Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. . Google Scholar

[18] Keller, M. and Lenz, D., Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math. Model. Nat. Phenom. 5(2010), 198–224. https://doi.org/10.1051/mmnp/20105409 Google Scholar

[19] Telcs, A., The art of random walks. Lecture Notes in Mathematics, Vol. 1885, Springer-Verlag, Berlin, 2006. Google Scholar

[20] Udeigwe, L. C. and Ermentrout, G. B., Waves and patterns on regular graphs. SIAM J. Appl. Dyn. Syst. 14(2015), 1102–1129. https://doi.org/10.1137/140969488 Google Scholar

[21] Varopoulus, N. T., Hardy-Littlewood theory for semigroups. J. Funct. Anal. 83(1985), 240–260. https://doi.org/10.1016/0022-1236(85)90087-4 Google Scholar

[22] Weber, A., Analysis of physical Laplacian and the heat flow on a locally finite graph. J. Math. Anal. Appl. 370(2010), 146–158. https://doi.org/10.1016/j.jmaa.2010.04.044 Google Scholar