scholarly journals An asymptotic evaluation of heat kernel for short time

1996 ◽  
pp. 104-107
Author(s):  
J. A. Yan
Keyword(s):  
Heat Kernel ◽  
Asymptotic Evaluation ◽  
Short Time

scholarly journals SHORT TIME FULL ASYMPTOTIC EXPANSION OF HYPOELLIPTIC HEAT KERNEL AT THE CUT LOCUS

2017 ◽  
Vol 5 ◽  
Author(s):  
YUZURU INAHAMA ◽  
SETSUO TANIGUCHI
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Euclidean Space ◽  
Heat Kernel ◽  
Compact Manifold ◽  
Cut Locus ◽  
Finite Set ◽  
Rough Path ◽  
Hypoelliptic Heat Kernel ◽  
Short Time

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on a Euclidean space and a compact manifold. We study the ‘cut locus’ case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differential equation. Our main tools are S. Watanabe’s distributional Malliavin calculus and T. Lyons’ rough path theory.

Feynman-Kac formula: regularized trace and short-time asymptotics of the heat kernel

2008 ◽  
Author(s):  
S. A. Stepin ◽  
Piotr Kielanowski ◽  
Anatol Odzijewicz ◽  
Martin Schlichenmaier ◽  
Theodore Voronov
Keyword(s):  
Heat Kernel ◽  
Regularized Trace ◽  
Short Time Asymptotics ◽  
Time Asymptotics ◽  
Short Time

scholarly journals Feeling boundary by Brownian motion in a ball

2021 ◽  
pp. 1-34
Author(s):  
G. Serafin
Keyword(s):  
Brownian Motion ◽  
Heat Kernel ◽  
Rates Of Convergence ◽  
Half Line ◽  
Short Time Asymptotics ◽  
Time Asymptotics ◽  
Explicit Formulae ◽  
Short Time ◽  
Dirichlet Heat Kernel

We establish short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. So far, such results were only known in simple cases where explicit formulae are available, i.e., for sets as half-line, interval and their products. Presented asymptotics may be considered as a complement or a generalization of the famous “principle of not feeling the boundary” in case of a ball. Following the metaphor, the principle reveals when the process does not feel the boundary, while we describe what happens when it starts feeling the boundary.

Heat Kernel Short-Time Expansion within the Scope of Feynman-Kac Formula

2010 ◽  
Author(s):  
S. A. Stepin ◽  
A. J. Rejrat ◽  
Piotr Kielanowski ◽  
Victor Buchstaber ◽  
Anatol Odzijewicz ◽  
...  
Keyword(s):  
Heat Kernel ◽  
Time Expansion ◽  
Short Time
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