Lecture notes on the DiPerna–Lions theory in abstract measure spaces
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, pp. 729-766.

These notes collect the lectures given by the first author in Toulouse, April 2014, on the well-posedness theory for continuity and transport equation in metric measure spaces, summarizing the joint work appeared in Analysis and PDE. The last part of the notes covers also more recent developments, due to the second author, on diffusion operators on metric measure spaces.

Ces notes résument les leçons données à Toulouse en avril 2014 par le premier auteur sur le caractère bien ou mal posé des problèmes liés aux équations de continuité et de transport dans les espaces métriques mesurés, reprenant un travail commun publié dans Analysis and PDE. La dernière partie des notes couvre également des développements plus récents, dus au second auteur, sur les opérateurs de diffusion dans les espaces métriques mesurés.

Published online:
DOI: 10.5802/afst.1551

Luigi Ambrosio 1; Dario Trevisan 2

1 Scuola Normale Superiore, Pisa, Italy
2 Università di Pisa, Italy
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AFST_2017_6_26_4_729_0,
     author = {Luigi Ambrosio and Dario Trevisan},
     title = {Lecture notes on the {DiPerna{\textendash}Lions} theory in abstract measure spaces},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {729--766},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {4},
     year = {2017},
     doi = {10.5802/afst.1551},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1551/}
}
TY  - JOUR
AU  - Luigi Ambrosio
AU  - Dario Trevisan
TI  - Lecture notes on the DiPerna–Lions theory in abstract measure spaces
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2017
SP  - 729
EP  - 766
VL  - 26
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1551/
DO  - 10.5802/afst.1551
LA  - en
ID  - AFST_2017_6_26_4_729_0
ER  - 
%0 Journal Article
%A Luigi Ambrosio
%A Dario Trevisan
%T Lecture notes on the DiPerna–Lions theory in abstract measure spaces
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2017
%P 729-766
%V 26
%N 4
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1551/
%R 10.5802/afst.1551
%G en
%F AFST_2017_6_26_4_729_0
Luigi Ambrosio; Dario Trevisan. Lecture notes on the DiPerna–Lions theory in abstract measure spaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 4, pp. 729-766. doi : 10.5802/afst.1551. https://afst.centre-mersenne.org/articles/10.5802/afst.1551/

[1] Sergio Albeverio; Shigeo Kusuoka Maximality of infinite-dimensional Dirichlet forms and Høegh-Krohn’s model of quantum fields, Ideas and methods in quantum and statistical physics (Oslo, 1988), Cambridge University Press, 1992, pp. 301-330 | Zbl

[2] Luigi Ambrosio Transport equation and Cauchy problem for BV vector fields, Invent. Math., Volume 158 (2004) no. 2, pp. 227-260 | DOI | Zbl

[3] Luigi Ambrosio Transport equation and Cauchy problem for non-smooth vector fields, Calculus of Variations and Non-Linear Partial Differential Equations (CIME Series, Cetraro, 2005) (Lecture Notes in Mathematics), Volume 1927, Springer, 2008, pp. 1-42 | Zbl

[4] Luigi Ambrosio; Gianluca Crippa Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, Transport equations and multi-D hyperbolic conservation laws (Lecture Notes of the Unione Matematica Italiana), Volume 5, Springer, 2008, pp. 3-57 | Zbl

[5] Luigi Ambrosio; Gianluca Crippa Continuity equations and ODE flows with non-smooth velocity, Proc. R. Soc. Edinb., Sect. A, Math., Volume 144 (2014) no. 6, pp. 1191-1244 | DOI | Zbl

[6] Luigi Ambrosio; Alessio Figalli On flows associated to Sobolev vector fields in Wiener spaces: an approach à la DiPerna-Lions, J. Funct. Anal., Volume 256 (2009) no. 1, pp. 179-214 | DOI | Zbl

[7] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2005, vii+333 pages | Zbl

[8] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., Volume 195 (2014) no. 2, pp. 289-391 | DOI | Zbl

[9] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., Volume 163 (2014) no. 7, pp. 1405-1490 | DOI | Zbl

[10] Luigi Ambrosio; Giuseppe Savaré; Lorenzo Zambotti Existence and stability for Fokker-Planck equations with log-concave reference measure, Probab. Theory Relat. Fields, Volume 145 (2009) no. 3-4, pp. 517-564 | DOI | Zbl

[11] Luigi Ambrosio; Dario Trevisan Well posedness of Lagrangian flows and continuity equations in metric measure spaces, Anal. PDE, Volume 7 (2014) no. 5, pp. 1179-1234 | DOI | Zbl

[12] Dominique Bakry On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, New trends in stochastic analysis (Charingworth, 1994) (World Sci. Publ.), River Edge, 1997, pp. 43-75

[13] Dominique Bakry; Ivan Gentil; Michel Ledoux Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2014, xx+552 pages | Zbl

[14] David Bate Structure of measures in Lipschitz differentiability spaces, J. Am. Math. Soc., Volume 28 (2015) no. 2, pp. 421-482 | DOI | Zbl

[15] Marco Biroli; Umberto Mosco A Saint-Venant principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl., Volume 169 (1995), pp. 125-181 | DOI | Zbl

[16] Vladimir I. Bogachev Differentiable measures and the Malliavin calculus, Mathematical Surveys and Monographs, 164, American Mathematical Society, 2010, xv+488 pages | Zbl

[17] Nicolas Bouleau; Francis Hirsch Dirichlet forms and analysis on Wiener space, de Gruyter Studies in Mathematics, 14, De Gruyter, 1991, x+325 pages | Zbl

[18] Italo Capuzzo Dolcetta; Benoît Perthame On some analogy between different approaches to first order PDE’s with non smooth coefficients, Adv. Math. Sci. Appl., Volume 6 (1996) no. 2, pp. 689-703 | Zbl

[19] Nicolas Depauw Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d’un hyperplan, C. R., Math., Acad. Sci. Paris, Volume 337 (2003) no. 4, pp. 249-252 | DOI | Zbl

[20] Simone Di Marino Sobolev and BV spaces on metric measure spaces via derivations and integration by parts (2014) (https://arxiv.org/abs/1409.5620)

[21] Ronald J. DiPerna; Pierre-Louis Lions Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989) no. 3, pp. 511-547 | DOI | Zbl

[22] Andreas Eberle Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators, Lecture Notes in Mathematics, 1718, Springer, 1999, viii+262 pages | Zbl

[23] Stewart N. Ethier; Thomas G. Kurtz Markov processes, Characterization and convergence, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1986, x+534 pages | Zbl

[24] Alessio Figalli Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients, J. Funct. Anal., Volume 254 (2008) no. 1, p. 109-53 | DOI | Zbl

[25] Masatoshi Fukushima; Yoichi Oshima; Masayoshi Takeda Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics, 19, De Gruyter, 2011, x+489 pages | Zbl

[26] Nicola Gigli Nonsmooth differential geometry – An approach tailored for spaces with Ricci curvature bounded from below (2014) (https://arxiv.org/abs/1407.0809)

[27] Claude Le Bris; Pierre-Louis Lions Renormalized solutions of some transport equations with partially W 1,1 velocities and applications, Ann. Mat. Pura Appl., Volume 183 (2004) no. 1, pp. 97-130 | DOI | Zbl

[28] Claude Le Bris; Pierre-Louis Lions Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Commun. Partial Differ. Equations, Volume 33 (2008) no. 7, pp. 1272-1317 | DOI | Zbl

[29] Zhi-Ming Ma; Michael Röckner Introduction to the theory of (non-symmetric) Dirichlet forms, Universitext, Springer, 1992, viii+209 pages | Zbl

[30] Andrea Schioppa Derivations and Alberti representations, Adv. Math., Volume 293 (2016), pp. 436-528 | DOI | Zbl

[31] Ralph E. Showalter Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, 49, American Mathematical Society, 1997, xi+278 pages | Zbl

[32] Stanislav Konstantinovich Smirnov Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents, St. Petersbg. Math. J., Volume 5 (1994) no. 4, pp. 841-867 | Zbl

[33] Wilhelm Stannat The theory of generalized Dirichlet forms and its applications in analysis and stochastics, Mem. Am. Math. Soc., Volume 678 (1999), pp. 1-101 | Zbl

[34] Peter Stollmann A dual characterization of length spaces with application to Dirichlet metric spaces, Stud. Math., Volume 198 (2010) no. 3, pp. 221-233 | DOI | Zbl

[35] Daniel W. Stroock; S. R. Srinivasa Varadhan Multidimensional diffusion processes, Classics in Mathematics, Springer, 2006, xii+338 pages | Zbl

[36] Karl-Theodor Sturm Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math., Volume 32 (1995) no. 2, pp. 275-312 | Zbl

[37] Dario Trevisan (in preparation)

[38] Dario Trevisan Well-posedness of Diffusion Processes in Metric Measure Spaces, Scuola Normale Superiore (Italy) (2014) (Ph. D. Thesis)

[39] Dario Trevisan Lagrangian flows driven by BV fields in Wiener spaces, Probab. Theory Relat. Fields, Volume 163 (2015) no. 1-2, pp. 123-147 | DOI | Zbl

[40] Dario Trevisan Well-posedness for multidimensional diffusion processes with weakly differentiable coefficients, Electron. J. Probab., Volume 21 (2016) (Paper No. 22, 41 p.) | DOI | Zbl

[41] Nik Weaver Lipschitz algebras, World Scientific, 1999, xiii+233 pages | Zbl

[42] Nik Weaver Lipschitz algebras and derivations. II: Exterior differentiation, J. Funct. Anal., Volume 178 (2000) no. 1, pp. 64-112 | DOI | Zbl

[43] Toshio Yamada; Shinzo Watanabe On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., Volume 11 (1971), pp. 155-167 | DOI | Zbl

Cited by Sources: