Brownian motion on Perelman’s almost Ricci-flat manifold

References

[1] L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), no. 7, 1405–1490. 10.1215/00127094-2681605Search in Google Scholar

[2] L. Ambrosio, N. Gigli and G. Savaré, Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab. 43 (2015), no. 1, 339–404. 10.1214/14-AOP907Search in Google Scholar

[3] D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités XIX (1983/84), Lecture Notes in Math. 1123, Springer, Berlin (1985), 177–206. 10.1007/BFb0075847Search in Google Scholar

[4] D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li–Yau parabolic inequality, Rev. Mat. Iberoam. 22 (2006), no. 2, 683–702. 10.4171/RMI/470Search in Google Scholar

[5] Y. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 207–210. 10.1007/978-3-642-59938-5_14Search in Google Scholar

[6] L.-J. Cheng and A. Thalmaier, Characterization of pinched Ricci curvature by functional inequalities, preprint (2016), https://arxiv.org/abs/1611.02160. 10.1007/s12220-017-9905-1Search in Google Scholar

[7] D. Cordero-Erausquin, R. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), no. 2, 219–257. 10.1007/s002220100160Search in Google Scholar

[8] K. D. Elworthy, Stochastic differential equations on manifolds, London Math. Soc. Lecture Note Ser. 70, Cambridge University Press, Cambridge 1982. 10.1017/CBO9781107325609Search in Google Scholar

[9] M. Erbar, K. Kuwada and K.-T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math. 201 (2015), no. 3, 993–1071. 10.1007/s00222-014-0563-7Search in Google Scholar

[10] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635–681. 10.1007/978-3-642-59938-5_13Search in Google Scholar

[11] R. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993), no. 1, 225–243. 10.4310/jdg/1214453430Search in Google Scholar

[12] R. Haslhofer and A. Naber, Characterizations of the Ricci flow, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 5, 1269–1302. 10.4171/JEMS/787Search in Google Scholar

[13] R. Haslhofer and A. Naber, Ricci curvature and Bochner formulas for martingales, Comm. Pure Appl. Math. 71 (2018), no. 6, 1074–1108. 10.1002/cpa.21736Search in Google Scholar

[14] E. P. Hsu, Stochastic analysis on manifolds, Grad. Stud. Math. 38, American Mathematical Society, Providence 2002. 10.1090/gsm/038Search in Google Scholar

[15] T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), no. 520. 10.1090/memo/0520Search in Google Scholar

[16] K. Itô, Stochastic parallel displacement, Probabilistic methods in differential equations, Lecture Notes in Math. 451, Springer, Berlin (1975), 1–7. 10.1007/BFb0068575Search in Google Scholar

[17] J. Lott, Optimal transport and Perelman’s reduced volume, Calc. Var. Partial Differential Equations 36 (2009), no. 1, 49–84. 10.1007/s00526-009-0223-8Search in Google Scholar

[18] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903–991. 10.4007/annals.2009.169.903Search in Google Scholar

[19] P. Malliavin, Stochastic analysis, Grundlehren Math. Wiss. 313, Springer, Berlin 1997. 10.1007/978-3-642-15074-6Search in Google Scholar

[20] A. Naber, Characterizations of bounded Ricci curvature on smooth and nonsmooth spaces, preprint (2013), https://arxiv.org/abs/1306.6512v1. Search in Google Scholar

[21] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2003), https://arxiv.org/abs/math/0303109. Search in Google Scholar

[22] M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), no. 7, 923–940. 10.1002/cpa.20060Search in Google Scholar

[23] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math. 22 (1969), 345–400. 10.1002/cpa.3160220304Search in Google Scholar

[24] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients II, Comm. Pure Appl. Math. 22 (1969), 479–530. 10.1002/cpa.3160220404Search in Google Scholar

[25] D. W. Stroock and S. R. S. Varadhan, Multidimensional diffusion processes. Reprint of the 1997 edition, Classics Math., Springer, Berlin 2006. 10.1007/3-540-28999-2Search in Google Scholar

[26] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), 65–131. 10.1007/s11511-006-0002-8Search in Google Scholar

[27] K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133–177. 10.1007/s11511-006-0003-7Search in Google Scholar

[28] T. Tao, Poincaré’s legacies, pages from year two of a mathematical blog. Part II, American Mathematical Society, Providence 2009. Search in Google Scholar

[29] F.-Y. Wang and B. Wu, Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds, preprint (2016), https://arxiv.org/abs/1605.02447. 10.1007/s11425-017-9296-8Search in Google Scholar

[30] G. Wei, Curvature formulas in Section 6.1 of Perelman’s paper, notes available at http://web.math.ucsb.edu/~wei/paper/perelman-note.pdfSearch in Google Scholar