Abstract
In this chapter, we will consider various inequalities which imply (or which are equivalent to) the Nash-type heat kernel upper bound, i.e. \(p_{t}(x,y) \leq c_{1}{t}^{-\theta /2}\) for some θ > 0. We will also discuss Poincaré inequalities and their relations to heat kernel estimates in Sect. 3.3. We will prefer to discuss them under a general framework including weighted graphs.
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Notes
- 1.
\(\mathcal{F}\) is a domain of the Dirichlet form. For example, when \(\mathcal{E}(f,f) = \frac{1} {2}\int _{{\mathbb{R}}^{d}}\vert \nabla f{\vert }^{2}dx\), then \(\mathcal{F} = {W}^{1,2}({\mathbb{R}}^{d})\), the classical Sobolev space. When we consider weighted graphs, we may take \(\mathcal{F} = {\mathbb{L}}^{2}(X,\mu )\) (or \(\mathcal{F} = {H}^{2}\)).
References
S. Andres, M.T. Barlow, Energy inequalities for cutoff-functions and some applications. J. Reine Angew. Math. (to appear)
M.T. Barlow, Which values of the volume growth and escape time exponent are possible for a graph? Rev. Mat. Iberoam. 20, 1–31 (2004)
M.T. Barlow, Random Walks on Graphs. Lecture Notes for RIMS Lectures in 2005. Available at http://www.math.kyoto-u.ac.jp/~kumagai/rim10.pdf
M.T. Barlow, Random Walks on Graphs (Cambridge University Press, to appear)
M.T. Barlow, R.F. Bass, Stability of parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356, 1501–1533 (2003)
M.T. Barlow, R.F. Bass, Z.-Q. Chen, M. Kassmann, Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. 361, 1963–1999 (2009)
M.T. Barlow, R.F. Bass, T. Kumagai, Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Jpn. 58, 485–519 (2006)
M.T. Barlow, A. Grigor’yan, T. Kumagai, On the equivalence of parabolic Harnack inequalities and heat kernel estimates. J. Math. Soc. Jpn. 64, 1091–1146 (2012)
B. Bollobás, Random Graphs, 2nd edn. (Cambridge University Press, Cambridge, 2001)
E.A. Carlen, S. Kusuoka, D.W. Stroock, Upper bounds for symmetric Markov transition functions. Ann. Inst. Henri Poincaré Probab. Stat. 23, 245–287 (1987)
Z.-Q. Chen, M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory (Princeton University Press, Princeton, 2011)
T. Coulhon, Heat kernel and isoperimetry on non-compact Riemannian manifolds, in Contemporary Mathematics, vol. 338 (American Mathematical Society, Providence, 2003), pp. 65–99
T. Coulhon, Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141, 510–539 (1996)
T. Delmotte, Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoam. 15, 181–232 (1999)
E.B. Fabes, D.W. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Ration. Mech. Anal. 96, 327–338 (1986)
M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes (de Gruyter, Berlin, 1994)
A. Grigor’yan, The heat equation on non-compact Riemannian manifolds. Matem. Sbornik. 182, 55–87 (1991, in Russian) (English transl.) Math. USSR Sb. 72, 47–77 (1992)
A. Grigor’yan, Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoam. 10, 395–452 (1994)
A. Grigor’yan, L. Saloff-Coste, Heat kernel on manifolds with ends. Ann. Inst. Fourier (Grenoble) 59, 1917–1997 (2009)
P. Hajłasz, P. Koskela, Sobolev met Poincaré. Mem. Amer. Math. Soc. 145(688), x+101 pp. (2000)
B.M. Hambly, T. Kumagai, Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries, in Fractal Geometry and Applications: A Jubilee of B. Mandelbrot, Proceedings of Symposia in Pure Mathematics, vol. 72, Part 2 (American Mathematical Society, Providence, 2004), pp. 233–260
D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53, 503–523 (1986)
F. Lust-Piquard, Lower bounds on \(\left \Vert {K}^{n}\right \Vert _{1\rightarrow \infty }\) for some contractions K of L 2(μ), with some applications to Markov operators. Math. Ann. 303, 699–712 (1995)
B. Morris, Y. Peres, Evolving sets, mixing and heat kernel bounds. Probab. Theory Relat. Fields 133, 245–266 (2005)
J. Nash, Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931–954 (1958)
L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Not. IMRN 2, 27–38 (1992)
L. Saloff-Coste, Aspects of Sobolev-Type Inequalities. London Mathematical Society Lecture Notes, vol. 289 (Cambridge University Press, Cambridge, 2002)
K.T. Sturm, Analysis on local Dirichlet spaces—III. The parabolic Harnack inequality. J. Math. Pure Appl. 75, 273–297 (1996)
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Kumagai, T. (2014). Heat Kernel Estimates: General Theory. In: Random Walks on Disordered Media and their Scaling Limits. Lecture Notes in Mathematics(), vol 2101. Springer, Cham. https://doi.org/10.1007/978-3-319-03152-1_3
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