Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $W$-entropy formulas and Langevin deformation of flows on Wasserstein
  space over Riemannian manifolds   We introduce Perelman's $W$-entropy and prove the $W$-entropy formula along
the geodesic flow on the $L^2$-Wasserstein space over compact Riemannian
manifolds equipped with Otto's Riemannian metric, which allows us to recapture
a previous result due to Lott and Villani on the displacement convexity of
$s{\rm Ent}+ns\log s$ on $P^\infty_2(M)$ over Riemannian manifolds with
non-negative Ricci curvature. 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 equation of the Witten Laplacian on the
underlying manifolds, we introduce the Langevin deformation of flows on the
Wasserstein space over Riemannian manifold, which interpolates the gradient
flow and the geodesic flow on the Wasserstein space over Riemannian manifolds,
and can be regarded as the potential flow of the compressible Euler equation
with damping on manifolds. We prove the existence, uniqueness and regularity of
the Langevin deformation on the Wasserstein space over the Euclidean space and
compact Riemannian manifolds, and prove the convergence of the Langevin
deformation for $c\rightarrow 0$ and $c\rightarrow \infty$ respectively. We
prove an analogue of the Perelman type $W$-entropy formula along the Langevin
deformation on the Wasserstein space on Riemannian manifolds. A rigidity
theorem is proved for the $W$-entropy for the geodesic flow, and a rigidity
model is also provided for the Langevin deformation on the Wasserstein space
over complete Riemannian manifolds with the $CD(0, m)$-condition.

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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: hep-th
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an