Diffusions on graphs, Poisson problems and spectral geometry

McDonald, Patrick; Meyers, Robert

doi:10.1090/S0002-9947-02-02973-2

Diffusions on graphs, Poisson problems and spectral geometry
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by Patrick McDonald and Robert Meyers PDF

Trans. Amer. Math. Soc. 354 (2002), 5111-5136 Request permission

Abstract:

We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. We associate to certain subgraphs (domains) a pair of sequences, each of which is invariant under the action of the automorphism group of the underlying graph. We prove that these invariants differ by an explicit combinatorial factor given by Stirling numbers of the first and second kind. We prove that for any domain with a natural weighting, these invariants determine the eigenvalues of the Laplace operator corresponding to eigenvectors with nonzero mean. As a specific example, we investigate the relationship between our invariants and heat content asymptotics, expressing both as special values of an analog of a spectral zeta function.

References

David Aldous, Applications of random walks on finite graphs, Selected Proceedings of the Sheffield Symposium on Applied Probability (Sheffield, 1989) IMS Lecture Notes Monogr. Ser., vol. 18, Inst. Math. Statist., Hayward, CA, 1991, pp. 12–26. MR 1193058, DOI 10.1214/lnms/1215459284
Norman Biggs, Algebraic potential theory on graphs, Bull. London Math. Soc. 29 (1997), no. 6, 641–682. MR 1468054, DOI 10.1112/S0024609397003305
M. van den Berg and Peter B. Gilkey, Heat content asymptotics of a Riemannian manifold with boundary, J. Funct. Anal. 120 (1994), no. 1, 48–71. MR 1262245, DOI 10.1006/jfan.1994.1022
Fan R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. MR 1421568
Persi Diaconis and Daniel Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab. 1 (1991), no. 1, 36–61. MR 1097463
Jozef Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), no. 2, 787–794. MR 743744, DOI 10.1090/S0002-9947-1984-0743744-X
Peter G. Doyle and J. Laurie Snell, Random walks and electric networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984. MR 920811
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Robin Forman, Difference operators, covering spaces and determinants, Topology 28 (1989), no. 4, 413–438. MR 1030985, DOI 10.1016/0040-9383(89)90003-7
Robin Forman, Determinants of Laplacians on graphs, Topology 32 (1993), no. 1, 35–46. MR 1204404, DOI 10.1016/0040-9383(93)90035-T
Peter Gerl, Random walks on graphs with a strong isoperimetric property, J. Theoret. Probab. 1 (1988), no. 2, 171–187. MR 938257, DOI 10.1007/BF01046933
R. Has’minskii, Probabilistic representations of the solutions of some differential equations, In: Proc. 6th All-Union Conf. On Theor. Prob. and Math. Stat. (Vilnius 1960) (1960). (Russian)
G. Kirchhoff, Über die Auflösung der Gleichungen auf Welche man beider Untersuchen der linearen Vertheilung galvanischer Ströme gefüft wird, Annalen der Physik und Chemie 72 (1847), 495–508.
Kimberly K. J. Kinateder and Patrick McDonald, Variational principles for average exit time moments for diffusions in Euclidean space, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2767–2772. MR 1600101, DOI 10.1090/S0002-9939-99-04843-1
K. K. J. Kinateder, Patrick McDonald, and David Miller, Exit time moments, boundary value problems, and the geometry of domains in Euclidean space, Probab. Theory Related Fields 111 (1998), no. 4, 469–487. MR 1641818, DOI 10.1007/s004400050174
P. McDonald, Isoperimetric conditions, Poisson problems and diffusions in Riemannian manifolds, Potential Analysis 16 (2002), 115–138.
Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
Max Zorn, Continuous groups and Schwarz’ lemma, Trans. Amer. Math. Soc. 46 (1939), 1–22. MR 53, DOI 10.1090/S0002-9947-1939-0000053-7
N. Th. Varopoulos, Isoperimetric inequalities and Markov chains, J. Funct. Anal. 63 (1985), no. 2, 215–239. MR 803093, DOI 10.1016/0022-1236(85)90086-2
Nicolas Th. Varopoulos, Brownian motion and random walks on manifolds, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 2, 243–269. MR 746500
D. Zwillinger et al., eds., CRC Standard Mathematical Tables and Formulae, 30th Edition, CRC, New York, 1996.