Gradient Flows in Metric Spaces and in the Spaces of Probability Measures, and Applications to Fokker-Planck Equations with Respect to Log-Concave Measures

 Access to full text

 Full (PDF)

 Full (DjVu)

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. Zbl0798.46055MR1190532
2. AMBROSIO, L. - FUSCO, N. - PALLARA, D., Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). Zbl0957.49001MR1857292
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). Zbl1090.35002MR2129498
4. AMBROSIO, L. - SAVARÉ, G., Gradient flows in spaces of probability measures. Handbook of Differential Equations. Evolutionary equations III, North Holland2007. MR2549368DOI10.1016/S1874-5717(07)80004-1
5. AMBROSIO, L. - SAVARÉ, G. - ZAMBOTTI, L., Existence and Stability for Fokker-Planck equations with log-concave reference measure. (2007), Submitted paper. MR2529438DOI10.1007/s00440-008-0177-3
6. BÉNILAN, P., Solutions intégrales d'équations d'évolution dans un espace de Banach. C.R.Acad.Sci. Paris Sér. A-B, 274, (1972). Zbl0246.47068MR300164
7. BOGACHEV, V. I., Gaussian measures. Mathematical Surveys and Monographs, 62, AMS (1998). MR1642391DOI10.1090/surv/062
8. BORELL, C., Convex set functions in d-space. Period. Math. Hungar., 6 (1975), 111-136. Zbl0307.28009MR404559DOI10.1007/BF02018814
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. Zbl1082.76105MR2209130DOI10.1007/s00205-005-0386-1
11. CEPA, E., Problème de Skorohod multivoque, Annals of Probability, 26 no. 2 (1998), 500-532. Zbl0937.34046MR1626174DOI10.1214/aop/1022855642
12. DEBUSSCHE, A. - ZAMBOTTI, L., Conservative Stochastic Cahn-Hilliard equation with reflection, to appear in Annals of Probability (2007). Zbl1130.60068MR2349572DOI10.1214/009117906000000773
13. DAL MASO, G., An introduction to $\mathrm{\Gamma }$-convergence. Birkhauser (1993). Zbl0816.49001MR1201152DOI10.1007/978-1-4612-0327-8
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). Zbl1012.35001MR1985790DOI10.1017/CBO9780511543210
15. DA PRATO, G. - ZABCZYK, J., Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes, n.229, Cambridge University Press (1996). Zbl0849.60052MR1417491DOI10.1017/CBO9780511662829
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. Zbl0465.47041MR636814
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. Zbl1055.60052MR2036490DOI10.1007/s00440-003-0307-x
19. FUKUSHIMA, M. - OSHIMA, Y. - TAKEDA, M., Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin-New York (1994). Zbl0838.31001MR1303354DOI10.1515/9783110889741
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). MR2228384DOI10.1007/11429579_2
21. FUNAKI, T. - OLLA, S., Fluctuations for $\nabla \varphi$ interface model on a wall. Stoch. Proc. and Appl, 94 (2001), 1-27. Zbl1055.60096MR1835843DOI10.1016/S0304-4149(00)00104-6
22. FUNAKI, T. - SPOHN, H., Motion by mean curvature from the Ginzburg-Landau $\nabla \varphi$ interface model. Comm. Math. Phys.185 (1997), 1-36. Zbl0884.58098MR1463032DOI10.1007/s002200050080
23. GIACOMIN, G. - OLLA, S. - SPOHN, H., Equilibrium fluctuations for $\nabla \varphi$ interface model, Ann. Probab.29 (2001), 1138-1172. Zbl1017.60100MR1872740DOI10.1214/aop/1015345600
24. JOHN, F., Partial differential equations. Springer (4th. ed.) (1970). MR261133
25. JORDAN, R. - KINDERLEHRER, D. - OTTO, F., The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal.29 (1998), 1-17. Zbl0915.35120MR1617171DOI10.1137/S0036141096303359
26. LYTCHAK, A., Open map theorem for metric spaces. St. Petersburg Math. J., 17 (2006), 477-491. Zbl1152.53033MR2167848DOI10.1090/S1061-0022-06-00916-2
27. MA, Z. M. - ROCKNER, M., Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext, Springer-Verlag (1992). Zbl0826.31001MR1214375DOI10.1007/978-3-642-77739-4
28. NUALART, D. - PARDOUX, E., White noise driven quasilinear SPDEs with reflection, Prob. Theory and Rel. Fields, 93 (1992), 77-89. Zbl0767.60055MR1172940DOI10.1007/BF01195389
29. MCCANN, R. J., A convexity principle for interacting gases. Adv. Math., 128, (1997) 153-179. Zbl0901.49012MR1451422DOI10.1006/aima.1997.1634
30. OTTO, F., The geometry of dissipative evolution equations: the porous medium equation. Comm. PDE, 26 (2001), 101-174. Zbl0984.35089MR1842429DOI10.1081/PDE-100002243
31. PERELMAN, G. and PETRUNIN, A., Quasigeodesics and gradient curves in Alexandrov spaces. Unpublished preprint (1994). MR2693118
32. REVUZ, D. - YOR, M., Continuous Martingales and Brownian Motion, Springer Verlag (1991). Zbl0731.60002MR1083357DOI10.1007/978-3-662-21726-9
33. SAVARÉ, G., Gradient flows and diffusion semigroups in metric spaces and lower curvature bounds. CRAS note (2007), in press. MR2344814DOI10.1016/j.crma.2007.06.018
34. SHEFFIELD, S., Random Surfaces, Asterisque, No. 304 (2005). MR2251117
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. Zbl0935.82546MR1226387DOI10.1007/BF01049962
37. STROOCK, D. W. - VARADHAN, S. R. S., Multidimensional diffusion processes. Springer Verlag, second ed (1997). Zbl0426.60069MR532498
38. TANAKA, H., Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J.9 (1979), 163-177. Zbl0423.60055MR529332
39. VILLANI, C., Topics in optimal transportation. Graduate Studies in Mathematics, 58 (2003), AMS. Zbl1106.90001MR1964483DOI10.1007/b12016
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. Zbl1009.60047MR1921014DOI10.1007/s004400200203
41. ZAMBOTTI, L., Integration by parts on $\delta$-Bessel Bridges, $\delta >3$, and related SPDEs, Annals of Probability, 31 no. 1 (2003), 323-348. Zbl1019.60062MR1959795DOI10.1214/aop/1046294313
42. ZAMBOTTI, L., Fluctuations for a $\nabla \varphi$interface model with repulsion from a wall, Prob. Theory and Rel. Fields, 129 no. 3 (2004), 315-339. Zbl1073.60099MR2128236DOI10.1007/s00440-004-0335-1
43. ZAMBOTTI, L., Convergence of approximations of monotone gradient systems, Journal of Evolution Equations, 6 no. 4 (2006), 601-619. Zbl1130.35140MR2267701DOI10.1007/s00028-006-0275-6