Horizontal diffusion in C¹ path space
by Marc Arnaudon, Kolehe Abdoulaye Coulibaly and Anton Thalmaier

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 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.

Séminaire de Probabilités XLIII, 73-94
Lecture Notes in Mathematics 2006
Springer, 2011

The paper is available here:

Marc Arnaudon
Kolehe Abdoulaye Coulibaly
Anton Thalmaier

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