XXV: 19, 220-233, LNM 1485 (1991)
ÉMERY, Michel;
MOKOBODZKI, Gabriel
Sur le barycentre d'une probabilité dans une variété (
Stochastic differential geometry)
In a manifold $V$ (endowed with a connection), convex functions and continuous martingales can be defined, but expectations cannot. This article proposes to define the mass-centre of a probability $\mu$ on $V$ as a whole set of points, consisting of all $x$ in $V$ such that $f(x)\le\mu(f)$ for all bounded, convex $f$ on $V$. If $V$ is small enough, it is shown that this is equivalent to demanding that there exists (on a suitable filtered probability space) a continuous martingale $X$ such that $X_0=x$ and $X_1$ has law $\mu$
Comment: The conjecture (due to Émery) at the bottom of page 232 has been disproved by Kendall (
J. London Math. Soc. 46, 1992), as pointed out in
2650Keywords: Martingales in manifolds,
Convex functionsNature: Original
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