Gradient Flows in Metric Spaces and in the Spaces of Probability Measures, and Applications to Fokker-Planck Equations with Respect to Log-Concave Measures
Ambrosio, Luigi
Bollettino dell'Unione Matematica Italiana, Tome 1 (2008), p. 223-240 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

A survey on the main results of the theory of gradient flows in metric spaces and in the Wasserstein space of probability measures obtained in [3] and [4], is presented.

Publié le : 2008-02-01
@article{BUMI_2008_9_1_1_223_0,
     author = {Luigi Ambrosio},
     title = {Gradient Flows in Metric Spaces and in the Spaces of Probability Measures, and Applications to Fokker-Planck Equations with Respect to Log-Concave Measures},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {1},
     year = {2008},
     pages = {223-240},
     zbl = {1210.28005},
     mrnumber = {2388005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2008_9_1_1_223_0}
}

Ambrosio, Luigi. Gradient Flows in Metric Spaces and in the Spaces of Probability Measures, and Applications to Fokker-Planck Equations with Respect to Log-Concave Measures. Bollettino dell'Unione Matematica Italiana, Tome 1 (2008) pp. 223-240. http://gdmltest.u-ga.fr/item/BUMI_2008_9_1_1_223_0/

[1] Albeverio, S. - Kusuoka, S., Maximality of infinite-dimensional Dirichlet forms and Hegh-Krohn's model of quantum fields. Ideas and methods in quantum and statistical physics (Oslo, 1988), Cambridge Univ. Press, Cambridge (2002), 301-330. | MR 1190532 | Zbl 0798.46055

[2] Ambrosio, L. - Fusco, N. - Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | MR 1857292 | Zbl 0957.49001

[3] Ambrosio, L. - Gigli, N. - Savaré, G., Gradient flows in metric spaces and in the spaces of probability measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, (2005). | MR 2129498 | Zbl 1090.35002

[4] Ambrosio, L. - Savaré, G., Gradient flows in spaces of probability measures. Handbook of Differential Equations. Evolutionary equations III, North Holland 2007. | MR 2549368

[5] Ambrosio, L. - Savaré, G. - Zambotti, L., Existence and Stability for Fokker-Planck equations with log-concave reference measure. (2007), Submitted paper. | MR 2529438

[6] Bénilan, P., Solutions intégrales d'équations d'évolution dans un espace de Banach. C.R.Acad.Sci. Paris Sér. A-B, 274, (1972). | MR 300164 | Zbl 0246.47068

[7] Bogachev, V. I., Gaussian measures. Mathematical Surveys and Monographs, 62, AMS (1998). | MR 1642391

[8] Borell, C., Convex set functions in d-space. Period. Math. Hungar., 6 (1975), 111-136. | MR 404559 | Zbl 0307.28009

[9] Brézis, H., Opérateurs maximaux monotones. North-Holland, Amsterdam, (1973).

[10] Carrillo, J. A. - Mccann, R. - Villani, C., Contraction in the 2-Wasserstein length space and thermalization of granular media. Arch. Rational Mech. Anal., 179 (2006), 217-263. | MR 2209130 | Zbl 1082.76105

[11] Cepa, E., Problème de Skorohod multivoque, Annals of Probability, 26 no. 2 (1998), 500-532. | MR 1626174 | Zbl 0937.34046

[12] Debussche, A. - Zambotti, L., Conservative Stochastic Cahn-Hilliard equation with reflection, to appear in Annals of Probability (2007). | MR 2349572 | Zbl 1130.60068

[13] Dal Maso, G., An introduction to Γ-convergence. Birkhauser (1993). | MR 1201152 | Zbl 0816.49001

[14] Da Prato, G. - Zabczyk, J., Second order partial differential equations in Hilbert spaces, London Mathematical Society Lecture Notes Series, 293, Cambridge University Press (2002). | MR 1985790 | Zbl 1012.35001

[15] Da Prato, G. - Zabczyk, J., Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes, n.229, Cambridge University Press (1996). | MR 1417491 | Zbl 0849.60052

[16] De Giorgi, E. - Marino, A. and Tosques, M., Problems of evolution in metric spaces and maximal decreasing curves. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8) 68 (1980), 180-187. | MR 636814 | Zbl 0465.47041

[17] Fang, S., Wasserstein space over the Wiener space. Preprint, 2007.

[18] Feyel, D. - Ustunel, A. S., Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space, Probab. Theory Relat. Fields, 128 (2004), 347-385. | MR 2036490 | Zbl 1055.60052

[19] Fukushima, M. - Oshima, Y. - Takeda, M., Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin-New York (1994). | MR 1303354 | Zbl 0838.31001

[20] Funaki, T., Stochastic Interface Models. In: Lectures on Probability Theory and Statistics, Ecole d'Eté de Probabilités de Saint-Flour XXXIII - 2003 (ed. J. Picard), 103-274, Lect. Notes Math., 1869, Springer (2005). | MR 2228384

[21] Funaki, T. - Olla, S., Fluctuations for ∇⁡ϕ interface model on a wall. Stoch. Proc. and Appl, 94 (2001), 1-27. | MR 1835843 | Zbl 1055.60096

[22] Funaki, T. - Spohn, H., Motion by mean curvature from the Ginzburg-Landau ∇⁡ϕ interface model. Comm. Math. Phys. 185 (1997), 1-36. | MR 1463032 | Zbl 0884.58098

[23] Giacomin, G. - Olla, S. - Spohn, H., Equilibrium fluctuations for ∇⁡ϕ interface model, Ann. Probab. 29 (2001), 1138-1172. | MR 1872740 | Zbl 1017.60100

[24] John, F., Partial differential equations. Springer (4th. ed.) (1970). | MR 261133

[25] Jordan, R. - Kinderlehrer, D. - Otto, F., The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998), 1-17. | MR 1617171 | Zbl 0915.35120

[26] Lytchak, A., Open map theorem for metric spaces. St. Petersburg Math. J., 17 (2006), 477-491. | MR 2167848 | Zbl 1152.53033

[27] Ma, Z. M. - Rockner, M., Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext, Springer-Verlag (1992). | MR 1214375 | Zbl 0826.31001

[28] Nualart, D. - Pardoux, E., White noise driven quasilinear SPDEs with reflection, Prob. Theory and Rel. Fields, 93 (1992), 77-89. | MR 1172940 | Zbl 0767.60055

[29] Mccann, R. J., A convexity principle for interacting gases. Adv. Math., 128, (1997) 153-179. | MR 1451422 | Zbl 0901.49012

[30] Otto, F., The geometry of dissipative evolution equations: the porous medium equation. Comm. PDE, 26 (2001), 101-174. | MR 1842429 | Zbl 0984.35089

[31] Perelman, G. and Petrunin, A., Quasigeodesics and gradient curves in Alexandrov spaces. Unpublished preprint (1994). | MR 2693118

[32] Revuz, D. - Yor, M., Continuous Martingales and Brownian Motion, Springer Verlag (1991). | MR 1083357 | Zbl 0731.60002

[33] Savaré, G., Gradient flows and diffusion semigroups in metric spaces and lower curvature bounds. CRAS note (2007), in press. | MR 2344814

[34] Sheffield, S., Random Surfaces, Asterisque, No. 304 (2005). | MR 2251117

[35] Skorohod, A. V., Stochastic equations for diffusions in a bounded region, Theory Probab. Appl.6 (1961), 264-274.

[36] Spohn, H., Interface motion in models with stochastic dynamics, J. Stat. Phys.71 (1993), 1081-1132. | MR 1226387 | Zbl 0935.82546

[37] Stroock, D. W. - Varadhan, S. R. S., Multidimensional diffusion processes. Springer Verlag, second ed (1997). | MR 532498 | Zbl 0426.60069

[38] Tanaka, H., Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J.9 (1979), 163-177. | MR 529332 | Zbl 0423.60055

[39] Villani, C., Topics in optimal transportation. Graduate Studies in Mathematics, 58 (2003), AMS. | MR 1964483 | Zbl 1106.90001

[40] Zambotti, L., Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probab. Theory Related Fields, 123 no. 4 (2002), 579-600. | MR 1921014 | Zbl 1009.60047

[41] Zambotti, L., Integration by parts on δ-Bessel Bridges, δ>3, and related SPDEs, Annals of Probability, 31 no. 1 (2003), 323-348. | MR 1959795 | Zbl 1019.60062

[42] Zambotti, L., Fluctuations for a ∇⁡ϕinterface model with repulsion from a wall, Prob. Theory and Rel. Fields, 129 no. 3 (2004), 315-339. | MR 2128236 | Zbl 1073.60099

[43] Zambotti, L., Convergence of approximations of monotone gradient systems, Journal of Evolution Equations, 6 no. 4 (2006), 601-619. | MR 2267701 | Zbl 1130.35140