Feeling boundary by Brownian motion in a ball

Article type: Research Article

Authors: Serafin, G.^{; *}

Affiliations: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Poland. E-mail: grzegorz.serafin@pwr.edu.pl

Correspondence: [*] Corresponding author. 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.

Keywords: Heat kernel, ball, asymptotics, Laplacian, Brownian motion

DOI: 10.3233/ASY-211734

Journal: Asymptotic Analysis, vol. 129, no. 3-4, pp. 379-412, 2022