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Horizontal diffusion in *C*¹ path space

by Marc Arnaudon, Kolehe Abdoulaye
Coulibaly and Anton Thalmaier

**Abstract**

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.
* Séminaire de Probabilités * XLIII, 73-94

Lecture Notes in Mathematics **2006**

Springer, 2011

The paper is available here:

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