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Summary: Horizontal diffusion in C1 path space
Marc Arnaudon1
, Kol´eh`e Abdoulaye Coulibaly1,2
, and Anton Thalmaier2
1 Laboratoire de Math´ematiques et Applications, CNRS: UMR 6086, Universit´e de Poitiers,
T´el´eport 2 - BP 30179, F86962 Futuroscope Chasseneuil Cedex, France
marc.arnaudon@math.univ-poitiers.fr
2 Unit´e de Recherche en Math´ematiques, FSTC, Universit´e du Luxembourg,
6, rue Richard Coudenhove-Kalergi, L1359 Luxembourg, Grand-Duchy of Luxembourg
abdoulaye.coulibaly@uni.lu
anton.thalmaier@uni.lu
Summary. We define horizontal diffusion in C1 path space over a Riemannian manifold and
prove its existence. If the metric on the manifold is developing under the forward Ricci flow,
horizontal diffusion along Brownian motion turns out to be length preserving. As application,
we prove contraction properties in the Monge-Kantorovich minimization problem for prob-
ability measures evolving along the heat flow. For constant rank diffusions, differentiating a
family of coupled diffusions gives a derivative process with a covariant derivative of finite
variation. This construction provides an alternative method to filtering out redundant noise.
Key words: Brownian motion, damped parallel transport, horizontal diffusion,
Monge-Kantorovich problem, Ricci curvature
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