Affiliation:
1. Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Poland. E-mail: grzegorz.serafin@pwr.edu.pl
Abstract
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.
Reference25 articles.
1. Asymptotic behaviour of the Bessel heat kernels;Bogus;Math. Z.,2019
2. Sharp estimates of transition probability density for Bessel process in half-line;Bogus;Potential Anal.,2015
3. A lower bound for the heat kernel;Cheeger;Comm. Pure Appl. Math.,1981
4. Heat conduction and the principle of not feeling the boundary;Ciesielski;Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.,1966
5. The equivalence of certain heat kernel and Green function bounds;Davies;J. Funct. Anal.,1987