A cost on paths of measures which induces the Fokker-Planck equation. (English) Zbl 1347.49043
Summary: In [J. Math. Pures Appl. (9) 97, No. 4, 318–390 (2012; Zbl 1242.49059)], J. Feng and T. Nguyen defined a cost on curves of measures which is finite exactly on the curves which solve a Fokker-Planck equation with \(L^2\) drift. In this paper, using ideas of D. Gomes and E. Valdinoci, we give a different construction of the cost of Feng and Nguyen.
MSC:
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 35Q84 | Fokker-Planck equations |
Citations:
Zbl 1242.49059References:
| [1] | Adams, S.; Dirr, N.; Peletier, M.; Zimmer, J., From the large deviation principle to the Wasserstein gradient flow: a new micromacro passage, Comm. Math. Phys., 307, 791-815 (2011) · Zbl 1267.60106 |
| [2] | Ambrosio, L.; Gigli, N.; Savaré, G., Gradient Flows (2005), Birkhäeuser: Birkhäeuser Basel |
| [3] | Ambrosio, L.; Gigli, N.; Savaré, G., Heat flow and calculus on metric measure spaces with Ricci curvature bounded below—the compact case, (Analysis and numerics of PDE’s. Analysis and numerics of PDE’s, Springer-INdAM Ser., vol. 4 (2013), Springer: Springer Milan), 63-115 · Zbl 1292.53029 |
| [4] | Ambrosio, L.; Savaré, G.; Zambotti, L., Existence and stability for Fokker-Planck equations with log-concave reference measures, Probab. Theory Related Fields, 145, 517-564 (2009) · Zbl 1235.60105 |
| [5] | Bernard, P.; Buffoni, B., Optimal mass transportation and Mather theory, J. Eur. Math. Soc., 9, 85-121 (2007) · Zbl 1241.49025 |
| [6] | Bessi, U., A time step approximation scheme for a viscous version of the Vlasov equation, Adv. Math., 266, 17-83 (2014) · Zbl 1298.49044 |
| [7] | Bogachev, V.; Da Prato, G.; Röckner, M., Uniqueness for solutions of Fokker-Planck equations on infinite-dimensional spaces, J. Evol. Equ., 10, 487-509 (2010) · Zbl 1239.34063 |
| [9] | Da Prato, G.; Flandoli, F.; Röckner, M., Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift, Partial Differ. Equ. Anal. Comput., 2, 121-145 (2014) · Zbl 1328.35015 |
| [10] | De Pascale, L.; Gelli, M. S.; Granieri, L., Minimal measures, one-dimensional currents and the Monge-Kantorovich problem, Calc. Var. Partial Differential Equations, 27, 363-388 (2006) |
| [12] | Feng, J.; Nguyen, T., Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, J. Math. Pures Appl., 97, 318-390 (2012) · Zbl 1242.49059 |
| [13] | Gomes, D.; Valdinoci, E., Entropy penalization method for Hamilton-Jacobi equations, Adv. Math., 215, 94-152 (2007) · Zbl 1119.70013 |
| [14] | Léonard, C., From the Schrödinger problem to the Monge-Kantorovich problem, J. Funct. Anal., 262, 1879-1920 (2012) · Zbl 1236.49088 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
