Short—time Asymptotics of the Heat Kernel of the Laplacian for a Multiply-connected Domain inR2with Robin Boundary Conditions

2001 ◽  
Vol 77 (1-2) ◽  
pp. 177-194 ◽  
Author(s):  
E.M.E. Zayed
Keyword(s):  
Boundary Conditions ◽  
Heat Kernel ◽  
Connected Domain ◽  
Robin Boundary Conditions ◽  
Multiply Connected Domain ◽  
Multiply Connected ◽  
Short Time Asymptotics ◽  
Time Asymptotics ◽  
Short Time ◽  
Robin Boundary

scholarly journals The wave equation approach to an inverse eigenvalue problem for an arbitrary multiply connected drum inℝ2with Robin boundary conditions

2001 ◽  
Vol 25 (11) ◽  
pp. 717-726 ◽  
Author(s):  
E. M. E. Zayed ◽  
I. H. Abdel-Halim
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Inverse Eigenvalue Problem ◽  
Robin Boundary Conditions ◽  
Two Dimensional ◽  
Equation Approach ◽  
Multiply Connected ◽  
Inverse Eigenvalue ◽  
Negative Laplacian ◽  
Robin Boundary

The spectral functionμˆ(t)=∑j=1∞exp(−itμj1/2), where{μj}j=1∞are the eigenvalues of the two-dimensional negative Laplacian, is studied for small|t|for a variety of domains, where−∞<t<∞andi=−1. The dependencies ofμˆ(t)on the connectivity of a domain and the Robin boundary conditions are analyzed. Particular attention is given to an arbitrary multiply-connected drum inℝ2together with Robin boundary conditions on its boundaries.

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.

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