On Li-Yau Heat Kernel Estimate

2021 ◽  
Vol 37 (8) ◽  
pp. 1205-1218
Author(s):  
Lu Li ◽  
Zhen Lei Zhang
2019 ◽  
Vol 2019 (757) ◽  
pp. 89-130 ◽  
Author(s):  
Paul Horn ◽  
Yong Lin ◽  
Shuang Liu ◽  
Shing-Tung Yau

AbstractStudying the heat semigroup, we prove Li–Yau-type estimates for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality {\mathrm{CDE}^{\prime}(n,0)}, which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs (that is, graphs satisfying {\mathrm{CDE}^{\prime}(n,0)}) also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities. As a consequence, we obtain that the dimension of the space of harmonic functions on graphs with polynomial growth is finite. In the Riemannian setting, this was originally a conjecture of Yau, which was proved in that context by Colding and Minicozzi. Under the assumption that a graph has positive curvature, we derive a Bonnet–Myers-type theorem. That is, we show the diameter of positively curved graphs is finite and bounded above in terms of the positive curvature. This is accomplished by proving some logarithmic Sobolev inequalities.


Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.


Author(s):  
Jean-Michel Bismut

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.


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