A proof of the main identity

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Heat Kernel ◽  
Heat Kernels

This chapter proves the formula that was stated in Chapter 6. It first states various estimates on the hypoelliptic heat kernels, which are valid for b ≥ 1 and then makes a natural rescaling on the coordinates parametrizing ̂X. Next, the chapter introduces a conjugation on the Clifford variables and shows that the norm of the term defining the conjugation can be adequately controlled. The chapter then introduces a conjugate ℒA,bX of another ℒA,bX and its associated heat kernel. Afterward, the chapter obtains the limit as b → +∞ of the rescaled heat kernel, thus establishing the formula in Chapter 6. After further computations, this chapter states a result on convergence of heat kernels.

Rough estimates on the scalar heat kernel

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Hypoelliptic Operator ◽  
Uniform Bounds ◽  
Standard Heat ◽  
Probabilistic Construction

This chapter establishes rough estimates on the heat kernel rb,tX for the scalar hypoelliptic operator AbX on X defined in the preceding chapter. By rough estimates, this chapter refers to just the uniform bounds on the heat kernel. The chapter also obtains corresponding bounds for the heat kernels associated with operators AbX and another AbX over ̂X. Moreover, it gives a probabilistic construction of the heat kernels. This chapter also explains the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X and proves that as b → 0, the heat kernel rb,tX converges to the standard heat kernel of X.

Heat Kernels of Lorentz Cones

1999 ◽  
Vol 42 (2) ◽  
pp. 169-173 ◽  
Author(s):  
Hongming Ding
Keyword(s):  
Heat Kernel ◽  
Explicit Formula ◽  
Lower Bounds ◽  
Heat Kernels ◽  
Upper And Lower Bounds ◽  
Symmetric Cones ◽  
Lorentz Cone

AbstractWe obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time t and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.

Poisson’s integral formula via heat kernels

Analysis ◽  
2019 ◽  
Vol 39 (2) ◽  
pp. 59-64
Author(s):  
Yoichi Miyazaki
Keyword(s):  
Fourier Transform ◽  
Heat Kernel ◽  
Half Space ◽  
Unbounded Domain ◽  
Harmonic Functions ◽  
Integral Formula ◽  
Heat Kernels ◽  
Transform Method ◽  
Fourier Transform Method ◽  
The Fourier Transform

Abstract We give another proof of Poisson’s integral formula for harmonic functions in a ball or a half space by using heat kernels with Green’s formula. We wish to emphasize that this method works well even for a half space, which is an unbounded domain; the functions involved are integrable, since the heat kernel decays rapidly. This method needs no trick such as the subordination identity, which is indispensable when applying the Fourier transform method for a half space.

scholarly journals Exponential coordinates and regularity of groupoid heat kernels

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Bing So
Keyword(s):  
Differential Operator ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Pseudo Differential Operator

AbstractWe prove that on an asymptotically Euclidean boundary groupoid, the heat kernel of the Laplacian is a smooth groupoid pseudo-differential operator.

scholarly journals Pointwise monotonicity of heat kernels

2021 ◽  
Author(s):  
Diego Alonso-Orán ◽  
Fernando Chamizo ◽  
Ángel D. Martínez ◽  
Albert Mas
Keyword(s):  
Maximum Principle ◽  
Heat Kernel ◽  
Elementary Proof ◽  
Special Functions ◽  
Monotonicity Property ◽  
Heat Kernels ◽  
Hyperbolic Spaces ◽  
Special Cases ◽  
Special Points ◽  
New Inequalities

AbstractIn this paper we present an elementary proof of a pointwise radial monotonicity property of heat kernels that is shared by the Euclidean spaces, spheres and hyperbolic spaces. The main result was discovered by Cheeger and Yau in 1981 and rediscovered in special cases during the last few years. It deals with the monotonicity of the heat kernel from special points on revolution hypersurfaces. Our proof hinges on a non straightforward but elementary application of the parabolic maximum principle. As a consequence of the monotonicity property, we derive new inequalities involving classical special functions.

scholarly journals The subelliptic heat kernel on the three-dimensional solvable Lie groups

2015 ◽  
Vol 27 (4) ◽  
Author(s):  
Fabrice Baudoin ◽  
Matthew Cecil
Keyword(s):  
Heat Kernel ◽  
Lie Groups ◽  
Three Dimensional ◽  
Heat Kernels ◽  
Heat Semigroup ◽  
Solvable Lie Groups ◽  
New Type ◽  
Gradient Bounds

AbstractWe study the subelliptic heat kernels of the CR three-dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three-dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gradient bounds using a new type of curvature-dimension inequality.

scholarly journals Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian components

2016 ◽  
Vol 2016 (711) ◽  
Author(s):  
Zhen-Qing Chen ◽  
Panki Kim ◽  
Renming Song
Keyword(s):  
Heat Kernel ◽  
Heat Kernels ◽  
Kernel Estimates ◽  
Brownian Motions ◽  
Heat Kernel Estimates ◽  
Dirichlet Heat Kernel

AbstractIn this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in

scholarly journals Heat kernel asymptotic expansions for the Heisenberg sub-Laplacian and the Grushin operator

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140943
Author(s):  
Der-Chen Chang ◽  
Yutian Li
Keyword(s):  
Stationary Phase ◽  
Heisenberg Group ◽  
Heat Kernel ◽  
Asymptotic Expansions ◽  
Small Time ◽  
Heat Kernels ◽  
Grushin Operator ◽  
Geodesic Structure ◽  
Laplace Type ◽  
Method Of Steepest Descent

The sub-Laplacian on the Heisenberg group and the Grushin operator are typical examples of sub-elliptic operators. Their heat kernels are both given in the form of Laplace-type integrals. By using Laplace's method, the method of stationary phase and the method of steepest descent, we derive the small-time asymptotic expansions for these heat kernels, which are related to the geodesic structure of the induced geometries.

scholarly journals A HAMILTONIAN APPROACH TO THE HEAT KERNEL OF A SUBLAPLACIAN ON S2n+1

2013 ◽  
Vol 11 (06) ◽  
pp. 1350035 ◽  
Author(s):  
PETER C. GREINER
Keyword(s):  
Heat Kernel ◽  
Heat Kernels ◽  
Hamiltonian Approach ◽  
Subelliptic Operators ◽  
Cauchy Riemann ◽  
Classical Derivation

The heat kernel for the Cauchy–Riemann subLaplacian on S2n+1 is derived in a manner which is completely analogous to the classical derivation of elliptic heat kernels. This suggests that the classical Hamiltonian construction of elliptic heat kernels, with appropriate modifications, will yield heat kernels for subelliptic operators.

scholarly journals Coverings, Heat Kernels and Spanning Trees

10.37236/1444 ◽  
1998 ◽  
Vol 6 (1) ◽  
Author(s):  
Fan Chung ◽  
S.-T. Yau
Keyword(s):  
Heat Kernel ◽  
Regular Graph ◽  
Spanning Trees ◽  
Constant Factor ◽  
Heat Kernels ◽  
Regular Graphs ◽  
Regular Tree

We consider a graph $G$ and a covering $\tilde{G}$ of $G$ and we study the relations of their eigenvalues and heat kernels. We evaluate the heat kernel for an infinite $k$-regular tree and we examine the heat kernels for general $k$-regular graphs. In particular, we show that a $k$-regular graph on $n$ vertices has at most $$ (1+o(1)) {{2\log n}\over {kn \log k}} \left( {{ (k-1)^{k-1}}\over {(k^2-2k)^{k/2-1}}}\right)^n $$ spanning trees, which is best possible within a constant factor.

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