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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 Serie 9 1 (2008), fasc. n.1, p. 223-240, (English)
pdf (443 Kb), djvu (173 Kb). | MR 2388005 | Zbl 1210.28005

Sunto

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.
Referenze Bibliografiche
[1] S. ALBEVERIO - S. KUSUOKA, 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] L. AMBROSIO - N. FUSCO - D. PALLARA, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | MR 1857292 | Zbl 0957.49001
[3] L. AMBROSIO - N. GIGLI - G. SAVARÉ, 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] L. AMBROSIO - G. SAVARÉ, Gradient flows in spaces of probability measures. Handbook of Differential Equations. Evolutionary equations III, North Holland 2007. | fulltext (doi) | MR 2549368
[5] L. AMBROSIO - G. SAVARÉ - L. ZAMBOTTI, Existence and Stability for Fokker-Planck equations with log-concave reference measure. (2007), Submitted paper. | fulltext (doi) | MR 2529438
[6] P. BÉNILAN, 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] V. I. BOGACHEV, Gaussian measures. Mathematical Surveys and Monographs, 62, AMS (1998). | fulltext (doi) | MR 1642391
[8] C. BORELL, Convex set functions in d-space. Period. Math. Hungar., 6 (1975), 111-136. | fulltext (doi) | MR 404559 | Zbl 0307.28009
[9] H. BRÉZIS, Opérateurs maximaux monotones. North-Holland, Amsterdam, (1973).
[10] J. A. CARRILLO - R. MCCANN - C. VILLANI, Contraction in the 2-Wasserstein length space and thermalization of granular media. Arch. Rational Mech. Anal., 179 (2006), 217-263. | fulltext (doi) | MR 2209130 | Zbl 1082.76105
[11] E. CEPA, Problème de Skorohod multivoque, Annals of Probability, 26 no. 2 (1998), 500-532. | fulltext (doi) | MR 1626174 | Zbl 0937.34046
[12] A. DEBUSSCHE - L. ZAMBOTTI, Conservative Stochastic Cahn-Hilliard equation with reflection, to appear in Annals of Probability (2007). | fulltext (doi) | MR 2349572 | Zbl 1130.60068
[13] G. DAL MASO, An introduction to $\Gamma$-convergence. Birkhauser (1993). | fulltext (doi) | MR 1201152 | Zbl 0816.49001
[14] G. DA PRATO - J. ZABCZYK, Second order partial differential equations in Hilbert spaces, London Mathematical Society Lecture Notes Series, 293, Cambridge University Press (2002). | fulltext (doi) | MR 1985790 | Zbl 1012.35001
[15] G. DA PRATO - J. ZABCZYK, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes, n.229, Cambridge University Press (1996). | fulltext (doi) | MR 1417491 | Zbl 0849.60052
[16] E. DE GIORGI - A. MARINO and M. TOSQUES, 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. | fulltext bdim | MR 636814 | Zbl 0465.47041
[17] S. FANG, Wasserstein space over the Wiener space. Preprint, 2007.
[18] D. FEYEL - A. S. USTUNEL, Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space, Probab. Theory Relat. Fields, 128 (2004), 347-385. | fulltext (doi) | MR 2036490 | Zbl 1055.60052
[19] M. FUKUSHIMA - Y. OSHIMA - M. TAKEDA, Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin-New York (1994). | fulltext (doi) | MR 1303354 | Zbl 0838.31001
[20] T. FUNAKI, 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). | fulltext (doi) | MR 2228384
[21] T. FUNAKI - S. OLLA, Fluctuations for $\nabla\phi$ interface model on a wall. Stoch. Proc. and Appl, 94 (2001), 1-27. | fulltext (doi) | MR 1835843 | Zbl 1055.60096
[22] T. FUNAKI - H. SPOHN, Motion by mean curvature from the Ginzburg-Landau $\nabla\phi$ interface model. Comm. Math. Phys. 185 (1997), 1-36. | fulltext (doi) | MR 1463032 | Zbl 0884.58098
[23] G. GIACOMIN - S. OLLA - H. SPOHN, Equilibrium fluctuations for $\nabla\phi$ interface model, Ann. Probab. 29 (2001), 1138-1172. | fulltext (doi) | MR 1872740 | Zbl 1017.60100
[24] F. JOHN, Partial differential equations. Springer (4th. ed.) (1970). | MR 261133
[25] R. JORDAN - D. KINDERLEHRER - F. OTTO, The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998), 1-17. | fulltext (doi) | MR 1617171 | Zbl 0915.35120
[26] A. LYTCHAK, Open map theorem for metric spaces. St. Petersburg Math. J., 17 (2006), 477-491. | fulltext (doi) | MR 2167848 | Zbl 1152.53033
[27] Z. M. MA - M. ROCKNER, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext, Springer-Verlag (1992). | fulltext (doi) | MR 1214375 | Zbl 0826.31001
[28] D. NUALART - E. PARDOUX, White noise driven quasilinear SPDEs with reflection, Prob. Theory and Rel. Fields, 93 (1992), 77-89. | fulltext (doi) | MR 1172940 | Zbl 0767.60055
[29] R. J. MCCANN, A convexity principle for interacting gases. Adv. Math., 128, (1997) 153-179. | fulltext (doi) | MR 1451422 | Zbl 0901.49012
[30] F. OTTO, The geometry of dissipative evolution equations: the porous medium equation. Comm. PDE, 26 (2001), 101-174. | fulltext (doi) | MR 1842429 | Zbl 0984.35089
[31] G. PERELMAN and A. PETRUNIN, Quasigeodesics and gradient curves in Alexandrov spaces. Unpublished preprint (1994). | MR 2693118
[32] D. REVUZ - M. YOR, Continuous Martingales and Brownian Motion, Springer Verlag (1991). | fulltext (doi) | MR 1083357 | Zbl 0731.60002
[33] G. SAVARÉ, Gradient flows and diffusion semigroups in metric spaces and lower curvature bounds. CRAS note (2007), in press. | fulltext (doi) | MR 2344814
[34] S. SHEFFIELD, Random Surfaces, Asterisque, No. 304 (2005). | MR 2251117
[35] A. V. SKOROHOD, Stochastic equations for diffusions in a bounded region, Theory Probab. Appl. 6 (1961), 264-274.
[36] H. SPOHN, Interface motion in models with stochastic dynamics, J. Stat. Phys. 71 (1993), 1081-1132. | fulltext (doi) | MR 1226387 | Zbl 0935.82546
[37] D. W. STROOCK - S. R. S. VARADHAN, Multidimensional diffusion processes. Springer Verlag, second ed (1997). | MR 532498 | Zbl 0426.60069
[38] H. TANAKA, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1979), 163-177. | MR 529332 | Zbl 0423.60055
[39] C. VILLANI, Topics in optimal transportation. Graduate Studies in Mathematics, 58 (2003), AMS. | fulltext (doi) | MR 1964483 | Zbl 1106.90001
[40] L. ZAMBOTTI, 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. | fulltext (doi) | MR 1921014 | Zbl 1009.60047
[41] L. ZAMBOTTI, Integration by parts on $\delta$-Bessel Bridges, $\delta > 3$, and related SPDEs, Annals of Probability, 31 no. 1 (2003), 323-348. | fulltext (doi) | MR 1959795 | Zbl 1019.60062
[42] L. ZAMBOTTI, Fluctuations for a $\nabla\phi$interface model with repulsion from a wall, Prob. Theory and Rel. Fields, 129 no. 3 (2004), 315-339. | fulltext (doi) | MR 2128236 | Zbl 1073.60099
[43] L. ZAMBOTTI, Convergence of approximations of monotone gradient systems, Journal of Evolution Equations, 6 no. 4 (2006), 601-619. | fulltext (doi) | MR 2267701 | Zbl 1130.35140

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