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[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[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[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[24] 
F. JOHN, 
Partial differential equations. 
Springer (4th. ed.) (
1970). | 
MR 261133[31] 
G. PERELMAN and 
A. PETRUNIN, 
Quasigeodesics and gradient curves in Alexandrov spaces. Unpublished preprint (
1994). | 
MR 2693118[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.
[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[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