Fundamental solution of the heat equation on an arbitrary Riemannian manifold

1987 ◽  
Vol 41 (5) ◽  
pp. 386-389 ◽  
Author(s):  
A. A. Grigor'yan
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Solution ◽  
Heat Equation

scholarly journals Kesten's bound for subexponential densities on the real line and its multi-dimensional analogues

2018 ◽  
Vol 50 (2) ◽  
pp. 373-395 ◽  
Author(s):  
Dmitri Finkelshtein ◽  
Pasha Tkachov
Keyword(s):  
Fundamental Solution ◽  
Heat Equation ◽  
Probability Density ◽  
Real Line ◽  
The Real ◽  
Probability Densities ◽  
Tail Asymptotic

Abstract We study the tail asymptotic of subexponential probability densities on the real line. Namely, we show that the n-fold convolution of a subexponential probability density on the real line is asymptotically equivalent to this density multiplied by n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular subexponential functions and use it to find an analogue of Kesten's bound for functions on ℝd. The results are applied to the study of the fundamental solution to a nonlocal heat equation.

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.

Asymptotic behaviour of the fundamental solution to ∂ u /∂ t = — ( — ∆) m u

1993 ◽  
Vol 441 (1912) ◽  
pp. 423-432 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Infinite Number ◽  
Fourier Integral ◽  
Number Of Zeros

An asymptotic expansion is derived for the Fourier integral f ^ ( x ) = 1 ( 2 π ) n / 2 ∫ R n exp ( − | q | 2 m + i x ⋅ q ) d q , x ε R n as | x | →∞, where m is a positive integer. From this, it is deduced that the fundamental solution to the ‘heat’ equation ∂ u / ∂ t = − ( − Δ ) m u has an infinite number of zeros tending to infinity.

The Proof of the Feynman–Kac Formula for Heat Equation on a Compact Riemannian Manifold

2003 ◽  
Vol 06 (02) ◽  
pp. 311-320 ◽  
Author(s):  
Oleg O. Obrezkov
Keyword(s):  
Riemannian Manifold ◽  
Heat Equation ◽  
Compact Riemannian Manifold ◽  
Type Formula ◽  
Chernoff Theorem ◽  
Common Lines ◽  
Full Proof

A full proof of the Feynman–Kac-type formula for heat equation on a compact Riemannian manifold is obtained using some ideas originating from the papers of Smolyanov, Truman, Weizsaecker and Wittich.1-3 In particular, the technique exploited in the paper has some common lines with Chernoff theorem, which is one of the basic points of the approach to the topics undertaken in the above-mentioned papers.

scholarly journals A Remark on the Heat Equation and Minimal Morse Functions on Tori and Spheres

2013 ◽  
Vol 9 (17) ◽  
pp. 11-20
Author(s):  
Carlos Cadavid ◽  
Juan Diego Vélez
Keyword(s):  
Riemannian Manifold ◽  
Heat Equation ◽  
Critical Points ◽  
Morse Function ◽  
Initial Condition ◽  
Morse Functions ◽  
Flat Tori

Let (M, g)be a compact, connected riemannian manifold that is homogeneous, i.e. each pair of pointsp, q∈M have isometric neighborhoods. This paper is a first step towards an understanding of the extent to which it is true that for each “generic” initial condition f0, the solution to∂f /∂t= ∆gf, f (·,0) =f0is such that for sufficiently larget, f(·, t) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M. In this paper we show that this is true for flat tori and round spheres in all dimensions.

scholarly journals The inversion of a transform related to the Laplace transform and to heat conduction

1964 ◽  
Vol 4 (1) ◽  
pp. 1-14 ◽  
Author(s):  
David V. Widder
Keyword(s):  
Heat Conduction ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Laplace Transform ◽  
Physical Interpretation ◽  
Integral Transform ◽  
The Laplace Transform ◽  
Image Position

In a recent paper [7] the author considered, among other things, the integral transform where is the fundamental solution of the heat equation There we gave a physical interpretation of the transform (1.1). Here we shall choose a slightly different interpretation, more convenient for our present purposes. If then u(O, t) = f(t). That is, the function f(t) defined by equation (1.1) is the temperature at the origin (x = 0) of an infinite bar along the x-axis t seconds after it was at a temperature defined by the equation .

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