Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. (English) Zbl 1405.60122
Summary: We study contraction for the kinetic Fokker-Planck operator on the torus. Solving the stochastic differential equation, we show contraction and therefore exponential convergence in the Monge-Kantorovich-Wasserstein \(\mathcal{W}_2\) distance. Finally, we investigate if such a coupling can be obtained by a co-adapted coupling, and show that then the bound must depend on the square root of the initial distance.
MSC:
| 60J60 | Diffusion processes |
| 35Q84 | Fokker-Planck equations |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
Keywords:
convergence to equilibrium; hypocoercivity; Monge-Kantorovich-Wasserstein \(\mathcal{W}_2\) distance; Fokker-Planck equation; torus; co-adapted couplingsReferences:
| [1] | F. Bolley; I. Gentil; A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal., 263, 2430-2457 (2012) · Zbl 1253.35183 · doi:10.1016/j.jfa.2012.07.007 |
| [2] | F. Bolley; A. Guillin; F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, M2AN Math. Model. Numer. Anal., 44, 867-884 (2010) · Zbl 1201.82029 · doi:10.1051/m2an/2010045 |
| [3] | K. Burdzy; W. S. Kendall, Efficient Markovian couplings: Examples and counterexamples, Ann. Appl. Probab., 10, 362-409 (2000) · Zbl 1054.60077 · doi:10.1214/aoap/1019487348 |
| [4] | M. Chen, Optimal Markovian couplings and applications, Acta Math. Sinica (N. S.), 10 (1994), 260-275; A Chinese summary appears in Acta Math. Sinica, 38 (1995), p575. · Zbl 0813.60068 |
| [5] | S. Gadat; L. Miclo, Spectral decompositions and \(\begin{document} \mathbb{L}^2\end{document} \)-operator norms of toy hypocoercive semi-groups, Kinet. Relat. Models, 6, 317-372 (2013) · Zbl 1262.35134 · doi:10.3934/krm.2013.6.317 |
| [6] | F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46, 349-359 (2006) · Zbl 1096.35019 |
| [7] | F. Hérau; F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171, 151-218 (2004) · Zbl 1139.82323 · doi:10.1007/s00205-003-0276-3 |
| [8] | R. Jordan; D. Kinderlehrer; F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29, 1-17 (1998) · Zbl 0915.35120 · doi:10.1137/S0036141096303359 |
| [9] | W. S. Kendall, Coupling, local times, immersions, Bernoulli, 21, 1014-1046 (2015) · Zbl 1332.60118 · doi:10.3150/14-BEJ596 |
| [10] | K. Kuwada, Characterization of maximal Markovian couplings for diffusion processes, Electron. J. Probab., 14, 633-662 (2009) · Zbl 1191.60080 · doi:10.1214/EJP.v14-634 |
| [11] | S. Mischler; C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation, Arch. Ration. Mech. Anal., 221, 677-723 (2016) · Zbl 1338.35430 · doi:10.1007/s00205-016-0972-4 |
| [12] | P. Mörters and Y. Peres, Brownian Motion, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, 2010, URL http://books.google.co.uk/books?id=e-TbA-dSrzYC. · Zbl 1243.60002 |
| [13] | C. Mouhot; L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19, 969-998 (2006) · Zbl 1169.82306 · doi:10.1088/0951-7715/19/4/011 |
| [14] | D. Talay, Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Process. Related Fields, 8 (2002), 163-198, Inhomogeneous random systems (Cergy-Pontoise, 2001). · Zbl 1011.60039 |
| [15] | C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. · Zbl 1197.35004 |
| [16] | C. Villani, Optimal Transport: Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. · Zbl 1156.53003 |
| [17] | R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, USA, 2001, URL https://books.google.co.uk/books?id=4cI5136OdoMC. · Zbl 1267.82001 |
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