On the Green function of the killed fractional Laplacian on the periodic domain

TL;DR

This paper provides a simple proof of the positivity, unimodality, and complete monotonicity of the Green function for the killed fractional Laplacian on a periodic domain, using probabilistic and analytical methods.

Contribution

It introduces a novel, straightforward proof technique for key properties of the Green function for the killed fractional Laplacian on periodic domains.

Findings

Green function is positive and unimodal

Green function is completely monotone on the positive domain

Proof relies on Jacobi triple product and contour integration

Abstract

We give a very simple proof of the positivity and unimodality of the Green function for the killed fractional Laplacian on the periodic domain. The argument relies on the Jacobi triple product and a probabilistic representation of the Green function. We also show by a contour integration that the Green function is completely monotone on the positive part of the periodic domain.