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

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.