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
\hbox{$\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$}
$\mathit{\mu}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathit{N}}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathit{N}}{\mathit{\delta}}_{{\mathit{x}}_{\mathit{k}}}$ 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.