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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 2650

Keywords: Martingales in manifolds, Convex functions

Nature: Original

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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 (

Keywords: Martingales in manifolds, Convex functions

Nature: Original

Retrieve article from Numdam