The log entropy formula and entropy power for -heat equation on Riemannian manifolds
Introduction
In the seminal paper [1], Perelman introduced the -entropy and proved its monotonicity along the conjugated heat equation coupled with Ricci flow, that is the following the -entropy formula holds: which plays an important role in the proof of the Poincaré conjecture. Later, many people study the entropy formulae for other geometric evolution equations, such as heat equation [2], weighted heat equation [3] and other nonlinear diffusion equations on Riemannian manifolds [4], [5], [6] etc., see a recent survey [7].
Inspired by Perelman’s work, Ye [8] introduced a new log entropy functional where is an adjusted constant, denotes the scalar curvature, is a smooth solution to Ricci flow and is a smooth positive solution to the backward evolution equation With Perelman’s -entropy formula, using of a minimizing procedure, Ye proved the monotonicity of log entropy. As applications, the uniform log Sobolev inequality and Sobolev inequality along the Ricci flow were derived in [9], [10]. In [11], J. Li gave a precise formula of the first variation for instead of inequality of the first variation in [8].
Motivated by Perelman [1], Ni [2], Li [3] and Ye [8], in [12], Wu introduced a log entropy functional on Riemannian manifold with the static metric where is a constant and satisfies the following heat-type equation and proved its monotonicity formula according to Ni’s -entropy monotonicity formula [2]. Moreover, he also showed the weighted version log entropy monotonicity formula with nonnegative -dimensional Bakry–Emery–Ricci curvature by using Li’s -entropy monotonicity formula [3].
In the second part of this paper, we study the concavity of entropy power, which is a powerful tool in information theory and probability theory, especially it can derive various functional inequalities. In his 1948 seminal paper, Shannon [13] proved the entropy power inequality for two independent random vectors and , where the entropy power is defined by In 1985, Costa [14] showed an enhanced version of entropy power inequality which is equivalent to the concavity of entropy power, that is where , are probability densities solving the heat equation on . In 2000, Villani [15] gave a brief and more direct proof with an exact error term, which established a strong connection between the concavity of entropy power and some identities of Bakry-Emery [16]. In [17], Savaré and Toscani extended to the Rényi entropy power and proved its concavity along the porous medium equation and fast diffusion equation.
In [18], the authors studied the -entropy power and proved the concavity of -entropy power along the -heat equation on or a closed Riemannian manifold with nonnegative Ricci curvature.
On the other hand, in recent preprints [19], [20], S. Li and X-.D. Li proved the concavity of Shannon entropy power for the heat equation associated with the Witten Laplacian and the concavity of Rényi entropy power for the porous medium equation on weighted Riemannian manifolds with or condition, and also on or super Ricci flows.
On the basis of content, this paper is divided into two parts, the first part is to obtain -log entropy formula for -heat equation and prove its monotonicity on compact Riemannian manifold with nonnegative Ricci curvature and Ricci curvature bounded; the second part extends the concavity of entropy power for -heat equation to weighted case. Finally, we find that there are certain close relationships between log-entropy, entropy power and -entropy.
Section snippets
Notations and main results
Let be a closed weighted -dimensional Riemannian manifold, , where is the Riemannian volume measure. The weighted -Laplacian acting on smooth function is defined by where is the weighted divergence operator, if , is called the weighted Laplacian or Witten Laplacian. The -Bakry–Emery–Ricci curvature associated with the triple is defined by
Entropy dissipation formulae for weighted -heat equation
Let be a closed weighted Riemannian manifold, the linearized operator acted on a smooth function is defined by where . In the particular case , the operator at point satisfies The weighted -Bochner formula (in [23]) on with respect to the weighted -Laplacian operator is Applying -Bochner formula (3.2), we can obtain two entropy dissipation
The -log entropy formula
In this section, we consider the -log entropy (2.3) and devote to prove its monotonicity along the -heat equation on -dimensional closed Riemannian manifold .
Proof of Theorem 2.1 Following the notations in the previous sections, -log entropy can be rewritten as where , and are the -Shannon entropy and -Fisher information in (3.3), (3.5), respectively. Then, the first order derivative of -log entropy is
The concavity of -Shannon entropy power
In [4], Kotschwar and Ni introduced the -entropy and proved the Perelman type -entropy monotonicity formula on Riemannian manifold with nonnegative Ricci curvature. The first author and coauthor generalized this result to the weighted Riemannian manifold with nonnegative -Bakry–Emery–Ricci curvature [21] and -Bakry–Emery–Ricci curvature bounded below [23], respectively. In [18], we have obtained the variational formula of -Shannon entropy power
The -entropy power, -log entropy and -W-entropy
In this section, we establish a connection between the entropy power, log entropy and -entropy. In [19], S. Li and X.-D. Li obtained the NIW formula between the Shannon entropy power , the Fisher information and the -entropy for the heat equation on complete Riemannian manifolds, where is the Shannon entropy, is the Fisher information, and -entropy is defined by In the same
Acknowledgements
This work has been partially supported by the National Natural Science Foundation of China, NSFC (Grant No. 11701347). The first author would like to thank Professor Xiang-Dong Li for his interest and illuminating discussion. The authors are also thankful to the anonymous reviewers and editors for their constructive comments and suggestions on the earlier version for this paper.
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A note on static spaces
2021, Results in PhysicsCitation Excerpt :There are several (partial) differential equations on (semi-) Riemannian manifolds that play a notable role in physics, such as Einstein field equations, Ricci flow, Schrödinger equation, Laplace equation, Poisson equation, heat equation, wave equation, (vacuum) static equation (see, e.g., [4,6,7,13,14,16,18,26,27,29,30,33,34]).
Ricci Vector Fields
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