  A stochastic algorithm is proposed, finding some elements from the set of
intrinsic $p$-mean(s) associated to a probability measure $\nu$ on a compact
Riemannian manifold and to $p\in[1,\infty)$. It is fed sequentially with
independent random variables $(Y_n)_{n\in \mathbb{N}}$ distributed according to
$\nu$, which is often the only available knowledge of $\nu$. Furthermore, the
algorithm is easy to implement, because it evolves like a Brownian motion
between the random times when it jumps in direction of one of the $Y_n$,
$n\in\mathbb{N}$. Its principle is based on simulated annealing and
homogenization, so that temperature and approximations schemes must be tuned up
(plus a regularizing scheme if $\nu$ does not admit a H\"{o}lderian density).
The analysis of the convergence is restricted to the case where the state space
is a circle. In its principle, the proof relies on the investigation of the
evolution of a time-inhomogeneous $\mathbb{L}^2$ functional and on the
corresponding spectral gap estimates due to Holley, Kusuoka and Stroock. But it
requires new estimates on the discrepancies between the unknown instantaneous
invariant measures and some convenient Gibbs measures.
