Means in complete manifolds: uniqueness and approximation

¹ Laboratoire de Mathématiques et Applications, CNRS: UMR 7348, Université de Poitiers, Téléport 2 – BP 30179, 86962 Futuroscope Chasseneuil Cedex, France
marc.arnaudon@math.univ-poitiers.fr
² Institut de Mathématique de Toulouse, CNRS: UMR 5219, 118, route de Narbonne, 31062 Toulouse Cedex 9, France
laurent.miclo@math.univ-toulouse.fr

Received: 13 July 2012
Revised: 26 November 2012

Abstract

Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x₁,...,x_N) ∈ M^N for Lebesgue measure in M^N, the measure has a unique p–mean e_p(x). As a consequence, if X = (X₁,...,X_N) is a M^N-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (X_n)_n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process e_p,n(ω) = e_p(X₁(ω),...,X_n(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.