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Heat Kernel Estimates: General Theory

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2101)

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.

Keywords

  • Heat Kernel
  • Weighted Graph
  • Isoperimetric Inequality
  • Dirichlet Form
  • Heat Kernel Estimate

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Fig. 3.1
Fig. 3.2

Notes

  1. 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 \(\mathcal{F} = {H}^{2}\)).

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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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