Geometry and analysis of Dirichlet forms. (English) Zbl 1253.53035
The interesting paper under review studies analytic and geometric properties of the Dirichlet forms. Precisely, let \(X\) be a locally compact, connected and separable Hausdorff space and \(m\) be a nonnegative Radon measure with support \(X.\) Consider a regular, strongly local Dirichlet form \(\mathcal E\) on \(L^{2}(X,m)\) and set \(d\) for the associated intrinsic distance. Assume that the topology induced by \(d\) coincides with the original topology on \(X\) and that \(X\) is compact, satisfies a doubling property and supports a weak \((1,2)\)-Poincaré inequality. The authors discuss the coincidence/non-coincidence of the intrinsic length structure and the gradient structure. Under the further assumption that the Ricci curvature of \(X\) is bounded from below in the sense of Lott-Sturm-Villani, the following assertions are shown to be equivalent:
(i) the heat flow of \(\mathcal E\) gives the unique gradient flow of \(\mathcal U_{\infty },\)
(ii) \(\mathcal E\) satisfies the Newtonian property,
(iii) the intrinsic length structure coincides with the gradient structure.
Moreover, for the standard (resistance) Dirichlet form on the Sierpinski gasket equipped with the Kusuoka measure, the authors identify the intrinsic length structure with the measurable Riemannian and the gradient structures. The above results are applied to the (coarse) Ricci curvatures and asymptotics of the gradient of the heat kernel.
(i) the heat flow of \(\mathcal E\) gives the unique gradient flow of \(\mathcal U_{\infty },\)
(ii) \(\mathcal E\) satisfies the Newtonian property,
(iii) the intrinsic length structure coincides with the gradient structure.
Moreover, for the standard (resistance) Dirichlet form on the Sierpinski gasket equipped with the Kusuoka measure, the authors identify the intrinsic length structure with the measurable Riemannian and the gradient structures. The above results are applied to the (coarse) Ricci curvatures and asymptotics of the gradient of the heat kernel.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35K05 | Heat equation |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |
Keywords:
Dirichlet form; intrinsic distance; length structure; differential structure; Sierpinski gasket; gradient flow; Ricci curvature; Poincaré inequality; metric measure spaceReferences:
| [1] | Ambrosio, L.; Gigli, N.; Savaré, G., (Gradient Flows in Metric Spaces and in the Space of Probability Measures. Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich (2008), Birkhäuser Verlag: Birkhäuser Verlag Basel), 334 · Zbl 1145.35001 |
| [2] | Ambrosio, L.; Gigli, N.; Savaré, G., Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below · Zbl 1312.53056 |
| [3] | Bakry, D., On sobolev and logarithmic inequalities for Markov semigroups, (New Trends in Stochastic Analysis. New Trends in Stochastic Analysis, Charingworth, 1994 (1997), World Scientific Publishing: World Scientific Publishing River Edge, NJ), 43-75 |
| [4] | Bakry, D.; Emery, M., Diffusions hypercontractives, (Seminaire de probabilities, vol. XIX (1983-1984)), 177-206 · Zbl 0561.60080 |
| [5] | Balogh, Z. M.; Engoulatov, A.; Hunziker, L.; Maasalo, O. E., Functional inequalities and Hamilton-Jacobi equations in geodesic spaces, Potential Anal., 36, 317-337 (2012) · Zbl 1330.70076 |
| [6] | Beurling, A.; Deny, J., Dirichlet spaces, Proc. Natl. Acad. Sci. USA, 45, 208-215 (1959) · Zbl 0089.08201 |
| [7] | Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9, 428-517 (1999) · Zbl 0942.58018 |
| [8] | Colding, T. H.; Minicozzi, W. P., Liouville theorems for harmonic sections and applications, Comm. Pure Appl. Math., 51, 113-138 (1998) · Zbl 0928.58022 |
| [9] | Erbar, M., The heat equation on manifolds as a gradient flow in the Wasserstein space, Ann. Inst. Henri Poincaré Probab. Statist., 46, 1-23 (2010) · Zbl 1215.35016 |
| [10] | R.L. Frank, D. Lenz, D. Wingert, Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. arXiv:1012.5050; R.L. Frank, D. Lenz, D. Wingert, Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. arXiv:1012.5050 · Zbl 1304.60081 |
| [11] | Fukushima, M.; Ōshima, Y.; Takeda, M., (Dirichlet Forms and Symmetric Markov Processes. Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics, vol. 19 (1994), Walter de Gruyter & Co.: Walter de Gruyter & Co. Berlin), 392 · Zbl 0838.31001 |
| [12] | Gigli, N., On the heat flow on metric measure spaces: existence, uniqueness and stability, Calc. Var. Partial Differential Equations, 39, 101-120 (2010) · Zbl 1200.35178 |
| [13] | N. Gigli, K. Kuwada, S.-I. Ohta, Heat flow on Alexandrov spaces. Preprint.; N. Gigli, K. Kuwada, S.-I. Ohta, Heat flow on Alexandrov spaces. Preprint. · Zbl 1267.58014 |
| [14] | Heinonen, J.; Koskela, P., Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181, 1-61 (1998) · Zbl 0915.30018 |
| [15] | Hino, M., Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals, Proc. Lond. Math. Soc., 100, 3, 269-302 (2010) · Zbl 1185.60089 |
| [16] | Hino, M.; Ramírez, J. A., Small-time Gaussian behavior of symmetric diffusion semigroups, Ann. Probab., 31, 1254-1295 (2003) · Zbl 1085.31008 |
| [17] | Jordan, R.; Kinderlehrer, D.; Otto, F., The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29, 1-17 (1998) · Zbl 0915.35120 |
| [18] | N. Juillet, Diffusion by optimal transport in Heisenberg group. Preprint.; N. Juillet, Diffusion by optimal transport in Heisenberg group. Preprint. · Zbl 1378.53044 |
| [19] | Kajino, N., Heat kernel asymptotics for measurable Riemannian structure on the Sierpinski gasket, Potential Anal., 36, 67-115 (2012) · Zbl 1248.28012 |
| [20] | Keith, S., Modulus and the Poincaré inequality on metric measure spaces, Math. Z., 245, 255-292 (2003) · Zbl 1037.31009 |
| [21] | Keith, S., Measurable differentiable structures and the Poincaré inequality, Indiana Univ. Math. J., 53, 1127-1150 (2004) · Zbl 1088.53030 |
| [22] | Kigami, J., Harmonic metric and Dirichlet form on the Sierpiński gasket, (Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals. Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals, Sanda, Kyoto, 1990. Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals. Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals, Sanda, Kyoto, 1990, Pitman Res. Notes Math. Ser., vol. 283 (1993), Longman Sci. Tech.: Longman Sci. Tech. Harlow), 201-218 · Zbl 0793.31005 |
| [23] | Kigami, J., (Analysis on Fractals. Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143 (2001), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0998.28004 |
| [24] | Kigami, J., Measurable Riemannian geometry on the Sierpínski gasket: the Kusuoka measure and the Gaussian heat kernel estimate, Math. Ann., 340, 781-804 (2008) · Zbl 1143.28004 |
| [25] | Koskela, P.; Shanmugalingam, N.; Tyson, J. T., Dirichlet forms, Poincaré inequalities, and the Sobolev spaces of Korevaar and Schoen, Potential Anal., 21, 241-262 (2004) · Zbl 1088.31005 |
| [26] | Kusuoka, S., Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst. Math. Sci., 25, 659-680 (1989) · Zbl 0694.60071 |
| [27] | Kuwada, K., Duality on gradient estimates and Wasserstein controls, J. Funct. Anal., 258, 3758-3774 (2010) · Zbl 1194.53032 |
| [28] | Lott, J.; Villani, C., Hamilton-Jacobi semigroup on length spaces and applications, J. Math. Pures Appl., 88, 9, 219-229 (2007) · Zbl 1210.53047 |
| [29] | Lott, J.; Villani, C., Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., 169, 2, 903-991 (2009) · Zbl 1178.53038 |
| [30] | Malliavin, P.; Stroock, D. W., Short time behavior of the heat kernel and its logarithmic derivatives, J. Differential Geom., 44, 550-570 (1996) · Zbl 0873.58063 |
| [31] | Norris, J. R., Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds, Acta Math., 179, 79-103 (1997) · Zbl 0912.58041 |
| [32] | Ohta, S.-I., Gradient flows on Wasserstein spaces over compact Alexandrov spaces, Amer. J. Math., 131, 475-516 (2009) · Zbl 1169.53053 |
| [33] | Ohta, S.-I.; Sturm, K.-T., Heat flow on Finsler manifolds, Comm. Pure Appl. Math., 62, 1386-1433 (2009) · Zbl 1176.58012 |
| [34] | Ollivier, Y., Ricci curvature of Markov chains on metric spaces, J. Funct. Anal., 256, 810-864 (2009) · Zbl 1181.53015 |
| [35] | Otto, F., The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26, 101-174 (2001) · Zbl 0984.35089 |
| [36] | Rajala, T., Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations, 44, 477-494 (2012) · Zbl 1250.53040 |
| [37] | T. Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm. Preprint.; T. Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm. Preprint. · Zbl 1260.53076 |
| [38] | Ramírez, J. A., Short-time asymptotics in Dirichlet spaces, Comm. Pure Appl. Math., 54, 259-293 (2001) · Zbl 1023.60049 |
| [39] | Savaré, G., Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Math. Acad. Sci. Paris, 345, 151-154 (2007) · Zbl 1125.53064 |
| [40] | Shanmugalingam, N., Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam., 16, 243-279 (2000) · Zbl 0974.46038 |
| [41] | Shanmugalingam, N., A universality property of Sobolev spaces in metric measure spaces, (Sobolev Spaces in Mathematics. I. Sobolev Spaces in Mathematics. I, Int. Math. Ser. (N. Y.), vol. 8 (2009), Springer: Springer New York), 345-359 · Zbl 1168.46306 |
| [42] | Stollmann, P., A dual characterization of length spaces with application to Dirichlet metric spaces, Studia Math., 198, 221-233 (2010) · Zbl 1198.31005 |
| [43] | Stroock, D. W.; Turetsky, J., Short time behavior of logarithmic derivatives of the heat kernel, Asian J. Math., 1, 17-33 (1997) · Zbl 0930.37027 |
| [44] | Sturm, K.-T., Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and \(L^p\)-Liouville properties, J. Reine Angew. Math., 456, 173-196 (1994) · Zbl 0806.53041 |
| [45] | Sturm, K.-T., Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math., 32, 275-312 (1995) · Zbl 0854.35015 |
| [46] | Sturm, K.-T., Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl., 75, 9, 273-297 (1996) · Zbl 0854.35016 |
| [47] | Sturm, K. T., Is a diffusion process determined by its intrinsic metric?, Chaos Solitons Fractals, 8, 1855-1860 (1997) · Zbl 0939.58032 |
| [48] | Sturm, K. T., The geometric aspect of Dirichlet forms, (New Directions in Dirichlet Forms. New Directions in Dirichlet Forms, AMS/IP Stud. Adv. Math., vol. 8 (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 233-277 · Zbl 0922.60074 |
| [49] | Sturm, K.-T., On the geometry of metric measure spaces. I, Acta Math., 196, 65-131 (2006) · Zbl 1105.53035 |
| [50] | Sturm, K.-T., On the geometry of metric measure spaces. II, Acta Math., 196, 133-177 (2006) · Zbl 1106.53032 |
| [51] | Villani, C., (Optimal Transport: Old and New. Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, vol. 338 (2009), Springer-Verlag: Springer-Verlag Berlin), 973 · Zbl 1156.53003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
