Heat kernel estimates for the Bessel differential operator in half-line

TL;DR

This paper derives sharp two-sided estimates for the heat kernel associated with the Bessel differential operator on the half-line, extending previous results to include the case rac{a0}{a0}0.

Contribution

It provides the first comprehensive sharp heat kernel estimates for the Bessel operator with rac{a0}{a0}0 on the half-line, complementing earlier work for rac{a0}{a0} eq 0.

Findings

Sharp two-sided heat kernel estimates for rac{a0}{a0}0 case

Extension of previous results to include rac{a0}{a0}0

Analytical and probabilistic methods used

Abstract

In the paper we consider the Bessel differential operator L^(\mu)=\dfrac{d^2}{dx^2}+\dfrac{2\mu+1}{x}\dfrac{d}{dx} in half-line (a,\infty), a>0, and its Dirichlet heat kernel p_a^(\mu)(t,x,y). For \mu=0, by combining analytical and probabilistic methods, we provide sharp two-sided estimates of the heat kernel for the whole range of the space parameters x,y>a and every t>0, which complements the recent results given in [1], where the case \mu\neq 0 was considered.