Geodesic
The max-plus Martin boundary
Documenta mathematica, Tome 14 (2009), pp. 195-240
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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.
DOI : 10.4171/dm/271
Classification : 31C35, 49L20
Mots-clés : potential theory, dynamic programming, eigenvalues, martin boundary, metric boundary, Lax-oleinik semigroup, weak KAM solutions, MAX-plus algebra, deterministic optimal control, Markov decision process, eigenvectors, Busemann functions, extremal generators 
@article{10_4171_dm_271,
     author = {Marianne Akian and St\'ephane Gaubert and Cormac Walsh},
     title = {The max-plus {Martin} boundary},
     journal = {Documenta mathematica},
     pages = {195--240},
     year = {2009},
     volume = {14},
     doi = {10.4171/dm/271},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/271/}
}
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Marianne Akian; Stéphane Gaubert; Cormac Walsh. The max-plus Martin boundary. Documenta mathematica, Tome 14 (2009), pp. 195-240. doi: 10.4171/dm/271

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