Asymptotic behaviour of the Bessel heat kernels

TL;DR

This paper investigates the long-term asymptotic behavior of Bessel heat kernels associated with a specific differential operator on a half-line, providing detailed expansions for certain parameter regimes.

Contribution

It offers new asymptotic expansions for the Dirichlet heat kernel of the Bessel operator in the half-line, extending understanding of its behavior for large spatial and temporal parameters.

Findings

Derived asymptotic expansions for the heat kernel as xy/t approaches infinity.

Provided detailed analysis of the kernel's behavior in the specified asymptotic regime.

Abstract

We consider Dirichlet heat kernel $p_a^{(\mu)}(t,x,y)$ for the Bessel differential operator $L^{(\mu)}=\frac{d^2}{dx^2}+\frac{2\mu+1}{2x}$, $\mu\in\mathbb{R}$, in half-line $(a,\infty)$, $a>0$, and provide its asymptotic expansions for $xy/t\rightarrow\infty$.