scholarly journals An asymptotic evaluation of heat kernel for short time

Author(s):  
J. A. Yan
2017 ◽  
Vol 5 ◽  
Author(s):  
YUZURU INAHAMA ◽  
SETSUO TANIGUCHI

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.


2008 ◽  
Author(s):  
S. A. Stepin ◽  
Piotr Kielanowski ◽  
Anatol Odzijewicz ◽  
Martin Schlichenmaier ◽  
Theodore Voronov

2021 ◽  
pp. 1-34
Author(s):  
G. Serafin

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.


2010 ◽  
Author(s):  
S. A. Stepin ◽  
A. J. Rejrat ◽  
Piotr Kielanowski ◽  
Victor Buchstaber ◽  
Anatol Odzijewicz ◽  
...  

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