The max-plus Martin boundary

TL;DR

This paper develops a max-plus potential theory framework, generalizing classical concepts like Martin boundaries to describe harmonic functions in deterministic control problems, with applications to eigenvector computation.

Contribution

It introduces a max-plus Martin boundary theory, extending potential theory to deterministic control, and provides a representation theorem for harmonic functions in this setting.

Findings

Max-plus Martin boundary generalizes classical boundary concepts.

Harmonic functions are represented via measures on the minimal boundary.

Eigenvectors of certain semigroups relate to Busemann points in normed spaces.

Abstract

We develop an idempotent version of probabilistic potential theory. The goal is to describe the set of max-plus harmonic functions, which give the stationary solutions of deterministic optimal control problems with additive reward. The analogue of the Martin compactification is seen to be a generalisation of the compactification of metric spaces using (generalised) Busemann functions. We define an analogue of the minimal Martin boundary and show that it can be identified with the set of limits of ``almost-geodesics'', and also the set of (normalised) harmonic functions that are extremal in the max-plus sense. Our main result is a max-plus analogue of the Martin representation theorem, which represents harmonic functions by measures supported on the minimal Martin boundary. We illustrate it by computing the eigenvectors of a class of translation invariant Lax-Oleinik semigroups. In this case, we relate the extremal eigenvectors to the Busemann points of a normed space.