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

scholarly journals SURVIVAL PROBABILITY (HEAT CONTENT) AND THE LOWEST EIGENVALUE OF DIRICHLET LAPLACIAN

2011 ◽  
Vol 25 (15) ◽  
pp. 1993-2007
Author(s):  
PAVOL KALINAY ◽  
LADISLAV ŠAMAJ ◽  
IGOR TRAVĚNEC
Keyword(s):  
Survival Probability ◽  
Heat Content ◽  
Simple Algorithm ◽  
Time Behavior ◽  
Dirichlet Laplacian ◽  
Geometric Characteristics ◽  
Long Time ◽  
Time Expansion ◽  
Short Time ◽  
Lowest Eigenvalue

We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this expansion, we look for a Padé interpolation between the short-time and the long-time behavior of the survival probability, i.e., between geometric characteristics of the boundary and the lowest eigenvalue of the Dirichlet Laplacian.

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.

scholarly journals Fast converging path integrals for time-dependent potentials: I. Recursive calculation of short-time expansion of the propagator

2011 ◽  
Vol 2011 (03) ◽  
pp. P03004 ◽  
Author(s):  
Antun Balaž ◽  
Ivana Vidanović ◽  
Aleksandar Bogojević ◽  
Aleksandar Belić ◽  
Axel Pelster
Keyword(s):  
Path Integrals ◽  
Time Dependent ◽  
Recursive Calculation ◽  
Time Expansion ◽  
Short Time
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