Feeling boundary by Brownian motion in a ball

Abstract

We provide short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. The boundary behaviour is precisely described. Presented results may be considered as a complement or a generalization of the famous "principle of not feeling the boundary" in case of a ball.

Laplace Dirichlet heat kernels in convex domains

“…Note that proper description of the exponential behaviour imposed the appearance of a new non-exponential factor h(t, x, y). Above estimates have been complemented with asymptotics in [22], which revealed that the behaviour of p B (t, x, y) is in fact driven by the expression δ x+y 2 / √ t. A similar property will be observed in general lower bound (5). We refer the reader to [3,4,5,12,19,20,21] for some other recent articles focused on sharp estimates of heat kernels in other settings.…”