Generalized displacement convexity for nonlinear mobility continuity equation and entropy power concavity on Wasserstein space over Riemannian manifolds. (English) Zbl 1521.49035
Summary: In this paper, we prove the generalized displacement convexity for nonlinear mobility continuity equation with \(p\)-Laplacian on Wasserstein space over Riemannian manifolds under the generalized McCann condition GMC\((m, n)\). Moreover, we obtain some variational formulae along the Langevin deformation of flows on the generalized Wasserstein space, which is the interpolation between the gradient flow and the geodesic flow. We also establish the connection between the displacement convexity of entropy functionals and the concavity of \(p\)-Rényi entropy powers. As an application, we derive the NIW formula which indicates the relationship between the \(p\)-Rényi entropy powers \({\mathcal{N}}_b\), the Fisher information \({\mathcal{I}}_b\) and the \({\mathcal{W}}\)-entropy.
MSC:
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 49K45 | Optimality conditions for problems involving randomness |
| 53C22 | Geodesics in global differential geometry |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 53E40 | Higher-order geometric flows |
References:
| [1] | Agueh, M., Finsler structure in the \(p\)-Wasserstein space and gradient flows, C. R. Acad. Sci. Paris, Ser., 350, 1, 35-40 (2012) · Zbl 1239.53021 · doi:10.1016/j.crma.2011.11.014 |
| [2] | Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics, Birkhauer (2005) · Zbl 1090.35002 |
| [3] | Benamou, J-D; Brenier, Y., A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84, 375-393 (2000) · Zbl 0968.76069 · doi:10.1007/s002110050002 |
| [4] | Brasco, L., A survey on dynamical transport distances, J. Math. Sci., 181, 6, 755-781 (2012) · Zbl 1348.49047 · doi:10.1007/s10958-012-0713-7 |
| [5] | Cardaliaguet, P.; Carlier, G.; Nazaret, B., Geodesics for a class of distances in the space of probability measures, Calc. Var. Partial Differential Equations, 48, 3-4, 395-420 (2013) · Zbl 1279.49029 · doi:10.1007/s00526-012-0555-7 |
| [6] | Carrillo, JA; Lisini, S.; Savaré, G.; Slepčev, D., Nonlinear mobility continuity equations and generalized displacement convexity, J. Funct. Anal., 258, 1273-1309 (2010) · Zbl 1225.49043 · doi:10.1016/j.jfa.2009.10.016 |
| [7] | Costa, M.: A new entropy power inequality. IEEE Trans. Inform. Theory IT-31, 751-760 (1985) · Zbl 0585.94006 |
| [8] | Daneri, S.; Savaré, G., Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 3, 1104-1122 (2008) · Zbl 1166.58011 · doi:10.1137/08071346X |
| [9] | Dolbeault, J.; Nazaret, B.; Savaré, G., A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34, 193-231 (2009) · Zbl 1157.49042 · doi:10.1007/s00526-008-0182-5 |
| [10] | Kotschwar, B.; Ni, L., Local gradient estimate for p-harmonic functions, \(1/H\) flow and an entropy formula, Ann. Sci. éc. Norm. Supér., 42, 1, 1-36 (2009) · Zbl 1182.53060 · doi:10.24033/asens.2089 |
| [11] | Lisini, S.; Marigonda, A., On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals, Manuscripta Math., 133, 1-2, 197-224 (2010) · Zbl 1196.49034 · doi:10.1007/s00229-010-0371-3 |
| [12] | Li, S.Z., Li, X.-D.: \(W\)-entropy formula and Langevin deformation of flows on Wasserstein space over Riemannian manifolds, arXiv:1604.02596 · Zbl 1400.58009 |
| [13] | Li, SZ; Li, X-D, \(W\)-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds, Sci. China Math., 61, 8, 1385-1406 (2018) · Zbl 1400.58009 · doi:10.1007/s11425-017-9227-7 |
| [14] | Li, S.Z., Li, X.-D.: On the Shannon entropy power on Riemannian manifolds and Ricci flows, arXiv:2001.00414v1, (2020) |
| [15] | Li, S.Z., Li, X.-D.: On Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on Riemannian manifolds, arXiv:2001.11184v1, (2020) |
| [16] | Li, S.Z., Li, X.-D.: On Shannon and Renyi entropy powers on Wasserstein space over Riemannian manifolds, preprint, (2021) |
| [17] | Li, X.-D., Wang, Y.-Z.: \(W\)-entropy formulae and entropy powers for \(p\)-Laplacian along geodesic flow on \(L^q\)-Wasserstein space over Riemannian manifolds, preprint, (2021) |
| [18] | Li, X-D, Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature, Math. Ann., 353, 2, 403-437 (2012) · Zbl 1280.58016 · doi:10.1007/s00208-011-0691-y |
| [19] | Lott, J., Optimal transport and Perelman’s reduced volume, Calc. Var., 36, 49-84 (2009) · Zbl 1171.53318 · doi:10.1007/s00526-009-0223-8 |
| [20] | Lott, J.; Villani, C., Ricci curvature for metric-measure spaces via optimal transport, Annals of Math., 169, 903-991 (2009) · Zbl 1178.53038 · doi:10.4007/annals.2009.169.903 |
| [21] | McCann, RJ, A convexity principle for interacting gases, Adv. Math., 128, 153-179 (1997) · Zbl 0901.49012 · doi:10.1006/aima.1997.1634 |
| [22] | Otto, F., The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differ. Equ., 26, 101-174 (2001) · Zbl 0984.35089 · doi:10.1081/PDE-100002243 |
| [23] | Savaré, G.; Toscani, G., The concavity of Rényi entropy power, IEEE Trans. Inform. Theory, 60, 5, 2687-2693 (2014) · Zbl 1360.94169 · doi:10.1109/TIT.2014.2309341 |
| [24] | Savaré, G.; Toscani, G., An information-theoretic proof of Nash’s inequality, Rend. Lincei Mat. Appl., 24, 83-93 (2013) · Zbl 1301.62006 |
| [25] | Shannon, CE, A mathematical theory of communication, Bell Syst. Tech. J., 27, 623-656 (1948) · Zbl 1154.94303 · doi:10.1002/j.1538-7305.1948.tb00917.x |
| [26] | Villani, C.: A short proof of the concavity of entropy power 46, 1695-1696 (2000) · Zbl 0994.94018 |
| [27] | Villani, C.: Topics in optimal transportation, Grad. Stud. Math., vol. 58, American Mathematical Society, Providence, RI, (2003) · Zbl 1106.90001 |
| [28] | Villani, C.: Optimal transport, old and new. Springer-Verlag, Berlin, Grundlehren der mathematischen Wissenschaften (2009) · Zbl 1156.53003 |
| [29] | Wang, Y-Z; Yang, J.; Chen, WY, Gradient estimates and entropy formulae for weighted p-heat equations on smooth metric, Acta Math, Sci. Ser. B Engl. Ed., 33, 4, 963-974 (2013) · Zbl 1299.58018 |
| [30] | Wang, Y-Z; Chen, WY, Gradient estimates and entropy formula for doubly nonlinear diffusion equations on Riemannian manifolds, Math. Methods Appl. Sci., 37, 2772-2781 (2014) · Zbl 1318.58014 · doi:10.1002/mma.3016 |
| [31] | Wang, Y-Z; Zhang, X-X, The concavity of p-entropy power and applications in functional inequalities, Nonlinear Anal., 179, 1-14 (2019) · Zbl 1404.35273 · doi:10.1016/j.na.2018.07.018 |
| [32] | Wang, Y.-Z., Wang, Y.-M.: The concavity of \(p\)-Rényi entropy power for doubly nonlinear diffusion equations and \(L^p\) -Gagliardo-Nirenberg-Sobolev inequalities, J. Math. Anal. Appl. 484 (1), 123698 (2020) · Zbl 1431.58014 |
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