The max-plus Martin boundary

  • Marianne Akian

    INRIA Saclay - hatIle-de- INRIA Saclay - hatIle-de- France and CMAP France and CMAP CMAP, École Polytechnique CMAP, École Polytechnique 91128 Palaiseau Cedex 91128 Palaiseau Cedex France France
  • Stéphane Gaubert

    INRIA Saclay - hatIle-de- INRIA Saclay - hatIle-de- France and CMAP France and CMAP CMAP, École Polytechnique CMAP, École Polytechnique 91128 Palaiseau Cedex 91128 Palaiseau Cedex France France
  • Cormac Walsh

    INRIA Saclay - hatIle-de- France and CMAP CMAP, École Polytechnique 91128 Palaiseau Cedex France
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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 Lax-Oleinik semigroups with nondifferentiable Lagrangian: we relate extremal eigenvector to Busemann points of normed spaces.

Cite this article

Marianne Akian, Stéphane Gaubert, Cormac Walsh, The max-plus Martin boundary. Doc. Math. 14 (2009), pp. 195–240

DOI 10.4171/DM/271