Volume 18, 2014
|Page(s)||185 - 206|
|Published online||27 March 2014|
Means in complete manifolds: uniqueness and approximation
1 Laboratoire de Mathématiques et
Applications, CNRS: UMR 7348, Université de Poitiers, Téléport 2 – BP
Futuroscope Chasseneuil Cedex,
2 Institut de Mathématique de Toulouse, CNRS: UMR 5219, 118, route de Narbonne, 31062 Toulouse Cedex 9, France
Revised: 26 November 2012
Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure has a unique p–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is well-defined. Assume M is compact and consider a probability measure ν in M. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν. When the set is a singleton, it converges to the p–mean.
Mathematics Subject Classification: 60D05 / 58C35 / 37A30 / 53C21 / 60J65
Key words: Stochastic algorithms / diffusion processes / simulated annealing / homogenization / probability measures on compact Riemannian manifolds / intrinsic p-means / instantaneous invariant measures / Gibbs measures / spectral gap at small temperature
© EDP Sciences, SMAI, 2014
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