scholarly journals On Groups, Slow Heat Kernel Decay Yields Liouville Property and Sharp Entropy Bounds

2018 ◽  
Author(s):  
Yuval Peres ◽  
Tianyi Zheng
Keyword(s):  
Heat Kernel ◽  
Liouville Property ◽  
Entropy Bounds ◽  
Heat Kernel Decay

scholarly journals Anomalous heat-kernel decay for random walk among bounded random conductances

2008 ◽  
Vol 44 (2) ◽  
pp. 374-392 ◽  
Author(s):  
N. Berger ◽  
M. Biskup ◽  
C. E. Hoffman ◽  
G. Kozma
Keyword(s):  
Random Walk ◽  
Heat Kernel ◽  
Heat Kernel Decay ◽  
Random Conductances ◽  
Anomalous Heat

scholarly journals Isoperimetry and heat kernel decay on percolation clusters

2004 ◽  
Vol 32 (1A) ◽  
pp. 100-128 ◽  
Author(s):  
Elisabeth Remy ◽  
Pierre Mathieu
Keyword(s):  
Heat Kernel ◽  
Percolation Clusters ◽  
Heat Kernel Decay

Manifolds and graphs with slow heat kernel decay

2001 ◽  
Vol 144 (3) ◽  
pp. 609-649 ◽  
Author(s):  
Martin Barlow ◽  
Thierry Coulhon ◽  
Alexander Grigor'yan
Keyword(s):  
Heat Kernel ◽  
Heat Kernel Decay

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

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.

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

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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