Horizontal diffusion in C¹ path space

2009/04/17 by Marc Arnaudon, Arnaudon, Marc, Abdoulaye Koléhè Coulibaly-Pasquier +3
Computer Science · Mathematics · #58J65 #60H30 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Probability (math.PR) #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.0904.2762

openalex publication_date 2009/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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