Asymptotic behaviour of the Bessel heat kernels

Abstract

We consider Dirichlet heat kernel p (μ) a (t, x, y) for the Bessel differential operator L (μ) = d 2 dx 2 + 2μ+1 2x , μ ∈ R, in half-line (a, ∞), a > 0, and provide its asymptotic expansions for x y/t → ∞.

Feeling boundary by Brownian motion in a ball

“…In particular, this implies |x − y| ≥ 1 2 δ(y). It is shown in the proof of Proposition 3 in [12] that for y ∈ B 15 16 x |x| , 1 16 the following estimate holds…”

“…Nevertheless, it still does not enable us to obtain precise asymptotics of the quotient k B (t, x, y)/k(t, x, y). See also [1,2,13,14,15,16,17] for other recent research on accurate exponential behaviour of heat kernels and densities of joint distribution of first hitting time and place. The goal of the paper is to derive uniform short-time asymptotics of the heat kernel k B (t, x, y) of the ball B(0, 1) as well as to provide rates of convergence.…”