Wasserstein gradient flows from large deviations of many-particle limits
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1166-1188.

We study the Fokker-Planck equation as the many-particle limit of a stochastic particle system on one hand and as a Wasserstein gradient flow on the other. We write the path-space rate functional, which characterises the large deviations from the expected trajectories, in such a way that the free energy appears explicitly. Next we use this formulation via the contraction principle to prove that the discrete time rate functional is asymptotically equivalent in the Gamma-convergence sense to the functional derived from the Wasserstein gradient discretization scheme.

DOI : 10.1051/cocv/2013049
Classification : 35A15, 5Q84
Mots-clés : Wasserstein, gradient flows, Fokker-Planck, gamma-convergence, large deviations 
@article{COCV_2013__19_4_1166_0,
     author = {Duong, Manh Hong and Laschos, Vaios and Renger, Michiel},
     title = {Wasserstein gradient flows from large deviations of many-particle limits},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1166--1188},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {4},
     year = {2013},
     doi = {10.1051/cocv/2013049},
     mrnumber = {3182684},
     zbl = {1284.35011},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2013049/}
}
TY  - JOUR
AU  - Duong, Manh Hong
AU  - Laschos, Vaios
AU  - Renger, Michiel
TI  - Wasserstein gradient flows from large deviations of many-particle limits
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 1166
EP  - 1188
VL  - 19
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2013049/
DO  - 10.1051/cocv/2013049
LA  - en
ID  - COCV_2013__19_4_1166_0
ER  - 
%0 Journal Article
%A Duong, Manh Hong
%A Laschos, Vaios
%A Renger, Michiel
%T Wasserstein gradient flows from large deviations of many-particle limits
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 1166-1188
%V 19
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2013049/
%R 10.1051/cocv/2013049
%G en
%F COCV_2013__19_4_1166_0
Duong, Manh Hong; Laschos, Vaios; Renger, Michiel. Wasserstein gradient flows from large deviations of many-particle limits. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1166-1188. doi : 10.1051/cocv/2013049. http://www.numdam.org/articles/10.1051/cocv/2013049/

[1] S. Adams, N. Dirr, M.A. Peletier and J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage. Commun. Math. Phys. 307 (2011) 791-815. | MR

[2] S. Adams, N. Dirr, M.A. Peletier and J. Zimmer, Large deviations and gradient flows. Philosophical Transactions of the Royal Society A. To appear (2013). | MR

[3] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. In Lect. Math., ETH Zürich. Birkhauser, Basel, 2nd edition (2008). | MR

[4] J.D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393. | MR

[5] A. Braides, Gamma convergence for beginners. Oxford University Press, Oxford (2002). | MR

[6] D.A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20 (1987) 247-308. | MR

[7] N. Dirr, V. Laschos and J. Zimmer, Upscaling from particle models to entropic gradient flows (submitted) (2010).

[8] R.M. Dudley, Real analysis and probability. Wadsworth and Brooks/Cole, Pacific Grove, CA, USA (1989). | MR

[9] A. Dembo and O. Zeitouni, Large deviations techniques and applications, in Stoch. Model. Appl. Probab., vol. 38. Springer, New York, NY, USA, 2nd edition (1987). | MR

[10] J. Feng and T.G. Kurtz, Large deviations for stochastic processes, in Mathematical surveys and monographs of vol. 131. AMS, Providence, RI, USA (2006). | MR | Zbl

[11] J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of convervations laws. J. Math. Pures Appl. 97 (2011) 318-390. | MR | Zbl

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

[13] C. Léonard, A large deviation approach to optimal transport. arxiv:org/abs/0710.1461v1 (2007).

[14] F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26 (2001) 101-174. | MR | Zbl

[15] M.A. Peletier, M. Renger and M. Veneroni, Variational formulation of the Fokker-Planck equation with decay: a particle approach. arxiv:org/abs/1108.3181 (2012). | MR | Zbl

[16] W. Rudin, Functional Analysis. McGraw-Hill, New York, NY, USA (1973). | MR

[17] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58. AMS, Providence (2003). | MR | Zbl

[18] C. Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer-Verlag, Berlin (2009). | MR | Zbl

Cité par Sources :