The<i>W</i>-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials

Li, Songzi; Li, Xiangdong

doi:10.2140/pjm.2015.278.173

Cited by 36 publications

(84 citation statements)

References 43 publications

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“…Our main results, Theorem 1.7 and Theorem 1.9, combine and extend two previous-hitherto unrelated-lines of developments: results in the setting of smooth families of time-dependent Riemannian manifolds that characterize solutions to Ric C 1 2 @ t g t 0 on I M (super-Ricci flows), e.g., by means of the monotonicity property (II) in terms of the L 2 -Wasserstein metric for the dual heat flow, initiated by work by Mc-Cann and Topping [39]; for subsequent work in this direction that also includes equivalences with gradient estimates (III) and coupling properties of backward Brownian motions; see, e.g., Topping [53], Philipowski-Kuwada [32,33], Arnaudon-Coulibaly-Thalmaier [9], Lakzian-Munn [34], Li-Li [35]. results for (static) metric measure spaces by Ambrosio-Gigli-Savare [6] as well as by Erbar-Kuwada-Sturm [17].…”

Section: Related Workmentioning

confidence: 99%

Heat Flow on Time‐Dependent Metric Measure Spaces and Super‐Ricci Flows

Kopfer

Sturm

2018

Comm Pure Appl Math

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We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness, and regularity. Of particular interest are properties that characterize the underlying space as a super-Ricci flow as previously introduced by the second author [51]. Our main result yields the equivalence of F dynamic convexity of the Boltzmann entropy on the (time-dependent) L 2 -Wasserstein space, F monotonicity of L 2 -Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time), F gradient estimates for the heat flow acting on functions (forward in time), and F a Bochner inequality involving the time-derivative of the metric. Moreover, we characterize the heat flow on functions as the unique forward EVI flow for the (time-dependent) energy in L 2 -Hilbert space and the dual heat flow on probability measures as the unique backward EVI flow for the (timedependent) Boltzmann entropy in L 2 -Wasserstein space.

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Section: Related Workmentioning

confidence: 99%

Heat Flow on Time‐Dependent Metric Measure Spaces and Super‐Ricci Flows

Kopfer

Sturm

2018

Comm Pure Appl Math

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“…The case of CD(K, m)-condition Theorem 2.1 can be viewed as the W -entropy formula for the heat equation of the Witten Laplacian on complete Riemannian manicolds with the CD(0, m)-condition. It is natural to raise the question whether we can extend Theorem 2.1 to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(K, m)-condition for general K ∈ R and m ∈ [n, ∞].In[26], we extended Theorem 2.1 to the Witten Laplacian on complete Riemannian manifolds with the CD(K, m)-condition for K ∈ R and m ∈ [n, ∞).…”

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confidence: 99%

“…[26,27]) Under the same notation as in Theorem 3.1, defineH m,K (u, t) = − M u log udµ − m 2 (1 + log(4πt)) − m Kt…”

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confidence: 99%

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

2018

Sci. China Math.

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In this survey paper, we give an overview of our recent works on the study of the W -entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W -entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W -entropy formula for the heat equation associated with the time dependent Witten Laplacian on n-dimensional compact manifolds equipped with a (K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the Wentropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W -entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the W -entropy formula for the Langevin deformation. Finally, we make a discussion on the W -entropy for the Ricci flow from the point of view of statistical mechanics and probability theory.MSC2010 Classification: primary 58J35, 58J65; secondary 60J60, 60H30.It is natural and interesting to ask the problems what is the hidden idea for Perelman to introduce the mysterious W -entropy, what is the reason for him to call the quantity in (1) the W -entropy, and whether there is some essential link between the W -entropy and the Boltzmann entropy in statistical mechanics and probability theory.Inspired by Perelman [44] and related works [39,40], the second author of this paper proved in [20] the W -entropy formula for the heat equation of the Witten Laplacian on compact Riemannian manifolds with the CD(0, m)-condition and gave a probabilistic interpretation of the W -entropy for the Ricci flow. Later, the W -entropy formula and a rigidity theorem for the W -entropy were proved in [22,24] for the fundamental solution to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(0, m)condition, and the W -entropy formula was proved in [21] for the Fokker-Planck equation of the Witten Laplacian on complete Riemannian manifolds with the CD(0, m)-condition. The relationship between Perelman's W -entropy formula for the Ricci flow and the Boltzmann H-theorem for the Boltzmann equation was discussed in [23] from the point of view of the statistical mechanics. In [26,27,29,30,31], we extended the W -entropy formula to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(K, m)condition and on compact Riemannian manifolds equipped with (K, m)-super Ricci flows. Moreover, we proved in [28,31] an analogue of the W -entropy formula for the geodesic flow on the Was...

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“…From [22,20,18,10,12,13], it has been known that there is a close connection between the differential Harnack inequality and the W -entropy for the heat equation on Riemannian manifolds. To see this link, let (M, g) be a complete Riemannian manifold with bounded geometry condition, u be a positive solution to the heat equation ∂ t u = ∆u.…”

Section: Introductionmentioning

confidence: 99%

“…Following [1], we say that (M, g, µ) satisfies the curvature-dimension CD(K, m)-condition 1 for a constant K ∈ R and m ∈ [n, ∞] if and only if See also our previous paper [13] and [14,15,16]. When φ is a constant and m = n, the (K, n)- (…”

Section: Introductionmentioning

confidence: 99%

TheW-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials

Cited by 36 publications

References 43 publications

Heat Flow on Time‐Dependent Metric Measure Spaces and Super‐Ricci Flows

Heat Flow on Time‐Dependent Metric Measure Spaces and Super‐Ricci Flows

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

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