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.

Refined estimates on the scalar heat kernel for bounded b

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Wave Equation ◽  
Heat Kernel ◽  
Distance Function ◽  
Uniform Bounds ◽  
Hypoelliptic Heat Kernel

This chapter obtains uniform bounds for the kernels rb,tX and another rb,tX for bounded b > 0, with the proper decay at infinity on X or ̂X. These bounds will be used to obtain corresponding bounds for the kernel qb,tX in the next chapter. Furthermore, the arguments developed in Chapter 12, which connect the hypoelliptic heat kernel with the wave equation, play an important role in proving the required estimates. Hence, this chapter first establishes estimates on the Hessian of the distance function on X. Then, the chapter obtains bounds on the first heat kernel rb,tX and establishes the bounds on another heat kernel rb,tX.

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.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

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.

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 A calculus approach to hyperfunctions III

1990 ◽  
Vol 118 ◽  
pp. 133-153 ◽  
Author(s):  
Tadato Matsuzawa
Keyword(s):  
Heat Equation ◽  
Heat Kernel ◽  
Compact Subset ◽  
Compact Support ◽  
Initial Values ◽  
Analytic Functionals

In the previous papers, [18] and [19], we have given some basis of a calculus approach to hyperfunctions. We have taken hyperfunctions with the compact support as initial values of the solutions of the heat equation. More precisely, let A′[K] be the space of analytic functionals supported by a compact subset K of Rn and let E(x, t) be the n-dimensional heat kernel given by.

scholarly journals Integral maximum principle and its applications

1994 ◽  
Vol 124 (2) ◽  
pp. 353-362 ◽  
Author(s):  
Alexander Grigor'yan
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Heat Equation ◽  
Heat Kernel ◽  
Integral Maximum Principle ◽  
Double Integrals

The integral maximum principle for the heat equation on a Riemannian manifold is improved and applied to obtain estimates of double integrals of the heat kernel.

scholarly journals ASYMPTOTIC EXPANSION FOR DAMPED WAVE EQUATIONS WITH PERIODIC COEFFICIENTS

2001 ◽  
Vol 11 (07) ◽  
pp. 1285-1310 ◽  
Author(s):  
R. ORIVE ◽  
E. ZUAZUA ◽  
A. F. PAZOTO
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Heat Kernel ◽  
Wave Equations ◽  
Bloch Wave ◽  
Periodic Coefficients ◽  
First Approximation ◽  
Wave Decomposition ◽  
Dissipative Wave Equation

We consider a linear dissipative wave equation in ℝN with periodic coefficients. By means of Bloch wave decomposition, we obtain an expansion of solutions as t→∞ and conclude that, in a first approximation, the solutions behave as the homogenized heat kernel.

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