On Properties of the Dirichlet Green's function for linear diffusions on a half line

TL;DR

This paper investigates the properties of Dirichlet Green's functions for one-dimensional linear diffusions on a half line, focusing on their ratio to the whole line Green's functions, including bounds, asymptotics, and concavity.

Contribution

It provides new bounds, asymptotic analysis, and a log concavity result for the ratio of Dirichlet to whole line Green's functions in this setting.

Findings

Derived bounds for the ratio of Green's functions.

Analyzed asymptotic behavior as diffusion coefficient approaches zero.

Proved a log concavity property of the ratio.

Abstract

This paper is concerned with the study of Green's functions for one dimensional diffusions with constant diffusion coefficient and linear time inhomogeneous drift. It is well know that the whole line Green's function is given by a Gaussian. Formulas for the Dirichlet Green's function on the half line are only known in special cases. The main object of study in the paper is the ratio of the Dirichlet to whole line Green's functions. Bounds, asymptotic behavior in the limit as the diffusion coefficient vanishes, and a log concavity result are obtained for this ratio.