Abstract
We consider heat kernels on different spaces such as Riemannian manifolds, graphs, and abstract metric measure spaces including fractals. The talk is an overview of the relationships between the heat kernel upper and lower bounds and the geometric properties of the underlying space. As an application, some estimate of higher eigenvalues of the Dirichlet problem is considered.
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References
G. K. Alexopoulos, A lower estimate for central probabilities on polycyclic groups, Can. J. Math., 44 (1992), no. 5, 897–910.
M. T. Barlow, Diffusions on fractals, in: Lectures on Probability Theory and Statistics, Ecole d’été de Probabilités de Saint-Flour XXV — 1995, Lecture Notes Math. 1690, Springer-Verlag, 1998. 1–121.
M. T. Barlow, R. F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math., 54 (1999), 673–744.
M. T. Barlow, A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Th. Rel. Fields, 79 (1988), 543–623.
P. Buser, A note on the isoperimetric constant, Ann. Scient. Ec. Norm. Sup., 15 (1982), 213–230.
I. Chavel, Eigenvalues in Riemannian geometry, Academic Press, New York, 1984.
I. Chavel, Isoperimetric inequalities and heat diffusion on Riemannian manifolds, Lecture notes 1999.
I. Chavel, E. A. FeldmanModified isoperimetric constants, and large time heat diffusion in Riemannian manifolds, Duke Math. J., 64 (1991), no. 3, 473–499.
F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics 92, AMS publications, 1996.
T. Coulhon, Heat kernels on non-compact Riemannian manifolds: a partial survey, Séminaire de théorie spectrale et géométrie, 1996–1997, Institut Fourier, Grenoble, 15 (1998), 167–187.
T. Coulhon, A. Grigor’yan, Ch. Pittet, A geometric approach to on-diagonal heat kernel lower bounds on groups, to appear in Ann. Inst. Fourier
T. Coulhon, L. Saloff-Coste, Isopérimétrie pour les groupes et les variétés, Revista Matematica Iberoamericana, 9 (1993), no. 2, 293–314.
E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, 1989.
E. B. Davies, Spectral theory and differential operators, Cambridge University Press, 1995.
T. Delmotte, Parabolic Harnack inequality and estimates of Markov chains on graphs, Revista Matematica Iberoamericana, 15 no. 1, (1999), 181–232.
M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and symmetric Markov processes, Studies in Mathematics 19, De Gruyter, 1994.
A. Grigor’yan, The heat equation on non-compact Riemannian manifolds, (in Russian) Matem. Sbornik, 182 (1991), no. 1, 55–87. Engl. transi. Math. USSR Sb., 72 (1992), no. 1, 47–77.
A. Grigor’yan, Heat kernel upper bounds on a complete non-compact manifold, Re-vista Matematica Iberoamericana, 10 (1994), no. 2, 395–452.
A. Grigor’yan, Heat kernel on a manifold with a local Harnack inequality, Comm Anal. Geom., 2 (1994), no. 1, 111–138.
A. Grigor’yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom., 45 (1997), 33–52.
A. Grigor’yan, Estimates of heat kernels on Riemannian manifolds, in: Spectral Theory and Geometry. ICMS Instructional Conference, Edinburgh 1998, ed. B.Davies and Yu.Safarov, London Math. Soc. Lecture Note Series 273, Cambridge Univ. Press, 1999. 140–225.
A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., 36 (1999), 135–249.
A. Grigor’yan, A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, preprint.
A. Grigor’yan, A. Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, to appear in Duke Math. J.
A. Grigor’yan, S.-T. Yau, On isoperimetric properties of higher eigenvalues of elliptic operators, in preparation.
W. Hebisch, L. Saloff Coste, Gaussian estimates for Markov chains and random walks on groups, Ann. Prob., 21 (1993), 673–709.
P. Li, S.-T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm Math. Phys., 88 (1983), 309–318.
P. Li, S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), no. 3–4, 153–201.
V. G. Maz’ya, Sobolev spaces,(in Russian) Izdat. Leningrad Gos. Univ. Leningrad, 1985. Engl. transi. Springer-Verlag, 1985.
Ch. Pittet, L. Saloff-Coste, A survey on the relationship between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples, preprint.
D. W. Robinson, Elliptic operators and Lie groups, Oxford Math. Mono., Clarenton Press, Oxford New York Tokyo, 1991.
L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Duke Math J., Internat. Math. Res. Notices, 2 (1992), 27–38.
L. Saloff-Coste, Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, 4 (1995), 429–467.
R. Schoen, S.-T. Yau, Lectures on Differential Geometry, Conference Proceedings and Lecture Notes in Geometry and Topology 1, International Press, 1994.
R. S. Strichartz, Analysis on fractals, Notices of AMS, 46 (1999), no. 10, 1199--1208.
D. W. Stroock, Estimates on the heat kernel for the second order divergence form operators, in: Probability theory. Proceedings of the 1989 Singapore Probability Conference held at the National University of Singapore, June 8–16 1989, ed. L.H.Y. Chen, K.P. Choi, K. Hu and J.H. Lou, Walter De Gruyter, 1992. 29–44.
K.-Th. Sturm, Analysis on local Dirichlet spaces III. The parabolic Harnack inequality, Journal de Mathématiques Pures et Appliquées, 75 (1996), no. 3, 273–297.
A. Telcs, Random walks on graphs, electric networks and fractals, J. Prob. Theo. and Rel. Fields, 82 (1989), 435–449.
V. I. Ushakov, Stabilization of solutions of the third mixed problem for a second order parabolic equation in a non-cylindric domain, (in Russian) Matem. Sbornik, 111 (1980), 95–115. Engl. transl. Math. USSR Sb., 39 (1981), 87–105.
N. Th. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and geometry on groups, Cambridge University Press, Cambridge, 1992.
W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138, Cambridge Univ. Press., 2000.
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Grigor’yan, A. (2001). Heat Kernels on Manifolds, Graphs and Fractals. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 201. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8268-2_22
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DOI: https://doi.org/10.1007/978-3-0348-8268-2_22
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