Spectral Analysis of Diffusions with Jump Boundary

TL;DR

This paper investigates the spectral properties of one-dimensional diffusions with jump boundaries, identifying a threshold drift where the spectral gap becomes independent of the drift, and addresses related elliptic eigenvalue problems.

Contribution

It introduces coupling methods to determine the spectral gap for large drifts and establishes a threshold beyond which the spectrum's bottom is unaffected by drift changes.

Findings

Identified a threshold drift where the spectral gap stabilizes.

Proved the spectral gap's independence from drift above the threshold.

Answered open questions on elliptic eigenvalue problems with non-local boundary conditions.

Abstract

In this paper we consider one-dimensional diffusions with constant coefficients in a finite interval with jump boundary and a certain deterministic jump distribution. We use coupling methods in order to identify the spectral gap in the case of a large drift and prove that that there is a threshold drift above which the bottom of the spectrum no longer depends on the drift. As a Corollary to our result we are able to answer two questions concerning elliptic eigenvalue problems with non-local boundary conditions formulated previously by Iddo Ben-Ari and Ross Pinsky.