Horizontal diffusion in $C^1$ path space

Marc Arnaudon (1) , Abdoulaye Koléhè Coulibaly-Pasquier (1) , Anton Thalmaier (2)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]
2 Institute of Mathematics

We define horizontal diffusion in $C^1$ 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 probability 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.

Mots clés: Brownian motion; damped parallel transport; horizontal diffusion; Monge-Kantorovich problem; Ricci curvature

Domaines: Probabilités [math.PR]

Marc Arnaudon, Abdoulaye Koléhè Coulibaly-Pasquier, Anton Thalmaier. Horizontal diffusion in $C^1$ path space. Séminaire de Probabilités, 2011, XLIII, pp.73-94. ⟨10.1007/978-3-642-15217-7_2⟩. ⟨hal-00376525⟩