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- Title
W-entropy and Langevin deformation on Wasserstein space over Riemannian manifolds.
- Authors
Li, Songzi; Li, Xiang-Dong
- Abstract
We prove the Perelman type W-entropy formula for the geodesic flow on the L 2 -Wasserstein space over a complete Riemannian manifold equipped with Otto's infinite dimensional Riemannian metric. To better understand the similarity between the W-entropy formula for the geodesic flow on the Wasserstein space and the W-entropy formula for the heat flow of the Witten Laplacian on the underlying manifold, we introduce the Langevin deformation of flows on the Wasserstein space over a Riemannian manifold, which interpolates the gradient flow and the geodesic flow on the Wasserstein space over a Riemannian manifold, and can be regarded as the potential flow of the compressible Euler equation with damping on a Riemannian manifold. We prove the existence, uniqueness and regularity of the Langevin deformation on the Wasserstein space over the Euclidean space and a compact Riemannian manifold, and prove the convergence of the Langevin deformation for c → 0 and c → ∞ respectively. Moreover, we prove the W-entropy-information formula along the Langevin deformation on the Wasserstein space on Riemannian manifolds. The rigidity theorems are proved for the W-entropy for the geodesic flow and the Langevin deformation on the Wasserstein space over complete Riemannian manifolds with the CD(0, m)-condition. Our results are new even in the case of Euclidean spaces and complete Riemannian manifolds with non-negative Ricci curvature.
- Subjects
GEODESIC flows; RIEMANNIAN manifolds; DEFORMATIONS (Mechanics); RIEMANNIAN metric; GEOMETRIC rigidity; POTENTIAL flow
- Publication
Probability Theory & Related Fields, 2024, Vol 188, Issue 3/4, p911
- ISSN
0178-8051
- Publication type
Article
- DOI
10.1007/s00440-023-01256-y