The max-plus Martin boundary

By Marianne Akian, Stephane Gaubert, Cormac Walsh. Documenta Mathematica, 2004

48 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3b8da5d193f25e28797c932074912fc71ef7e149

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Ricardo D. Katz
2. Stephane Gaubert
3. Cormac Walsh

In Computational Theory and Mathematics

1. Ricardo D. Katz
2. Stephane Gaubert
3. Cormac Walsh
4. Marianne Akian
5. Peter Butkovic

In Mathematical Physics

1. Ricardo D. Katz
2. Stephane Gaubert
3. Cormac Walsh

Citations in the corpus

• The Minkowski theorem for max-plus convex sets (2006, methodology): “called semimodules or idempotent linear spaces [LMS01, CGQ04]. Such structures have been used, for instance, to characterize the sets of stationary solutions of deterministic optimal control problems [AGW05], and to design numerical algorithms [FM00, AGL06], to mention a recent application. Max-plus”
• The horoboundary and isometry group of Thurston's Lipschitz metric (2010, background): “For more information, see [2], [21], and [1].”