Boundary traces of shift-invariant diffusions in half-plane

TL;DR

This paper characterizes the boundary traces of shift-invariant diffusions in the half-plane, showing they are Lévy processes with completely monotone jumps and establishing a bijective correspondence with such Lévy processes.

Contribution

It provides a complete characterization of boundary traces of shift-invariant diffusions, linking them to Lévy processes with completely monotone jumps and extending Krein's spectral theory.

Findings

Boundary traces are Lévy processes with completely monotone jumps.

Every Lévy process with completely monotone jumps arises as a boundary trace.

The correspondence between diffusions and Lévy processes is bijective up to transformations.

Abstract

We study boundary traces of shift-invariant diffusions: two-dimensional diffusions in the upper half-plane $\mathbb{R} \times [0, \infty)$ (or in $\mathbb{R} \times [0, R)$) invariant under horizontal translations. We prove that the corresponding trace processes are L\'evy processes with completely monotone jumps, and, conversely, every L\'evy process with completely monotone jumps is a boundary trace of some shift-invariant diffusion. Up to some natural transformations of space and time, this correspondence is bijective. We also reformulate this result in the language of additive functionals of the Brownian motion in $[0, \infty)$ (or in $[0, R)$), and Brownian excursions. Our main tool is the recent extension of Krein's spectral theory of strings, due to Eckhardt and Kostenko.