December  2018, 11(6): 1427-1441. doi: 10.3934/krm.2018056

Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

Received  April 2016 Revised  July 2017 Published  June 2018

Fund Project: The authors were supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis).

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.

Citation: Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
References:
[1]

F. BolleyI. Gentil and A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal., 263 (2012), 2430-2457.  doi: 10.1016/j.jfa.2012.07.007.

[2]

F. BolleyA. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, M2AN Math. Model. Numer. Anal., 44 (2010), 867-884.  doi: 10.1051/m2an/2010045.

[3]

K. Burdzy and W. S. Kendall, Efficient Markovian couplings: Examples and counterexamples, Ann. Appl. Probab., 10 (2000), 362-409.  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. doi: 10.1007/BF02560717.

[5]

S. Gadat and L. Miclo, Spectral decompositions and $ \mathbb{L}^2$-operator norms of toy hypocoercive semi-groups, Kinet. Relat. Models, 6 (2013), 317-372.  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 (2006), 349-359. 

[7]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.  doi: 10.1007/s00205-003-0276-3.

[8]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[9]

W. S. Kendall, Coupling, local times, immersions, Bernoulli, 21 (2015), 1014-1046.  doi: 10.3150/14-BEJ596.

[10]

K. Kuwada, Characterization of maximal Markovian couplings for diffusion processes, Electron. J. Probab., 14 (2009), 633-662.  doi: 10.1214/EJP.v14-634.

[11]

S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.  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. doi: 10.1017/CBO9780511750489.

[13]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  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).

[15]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5.

[16]

C. Villani, Optimal Transport: Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[17]

R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, USA, 2001, URL https://books.google.co.uk/books?id=4cI5136OdoMC.

show all references

References:
[1]

F. BolleyI. Gentil and A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal., 263 (2012), 2430-2457.  doi: 10.1016/j.jfa.2012.07.007.

[2]

F. BolleyA. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, M2AN Math. Model. Numer. Anal., 44 (2010), 867-884.  doi: 10.1051/m2an/2010045.

[3]

K. Burdzy and W. S. Kendall, Efficient Markovian couplings: Examples and counterexamples, Ann. Appl. Probab., 10 (2000), 362-409.  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. doi: 10.1007/BF02560717.

[5]

S. Gadat and L. Miclo, Spectral decompositions and $ \mathbb{L}^2$-operator norms of toy hypocoercive semi-groups, Kinet. Relat. Models, 6 (2013), 317-372.  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 (2006), 349-359. 

[7]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218.  doi: 10.1007/s00205-003-0276-3.

[8]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[9]

W. S. Kendall, Coupling, local times, immersions, Bernoulli, 21 (2015), 1014-1046.  doi: 10.3150/14-BEJ596.

[10]

K. Kuwada, Characterization of maximal Markovian couplings for diffusion processes, Electron. J. Probab., 14 (2009), 633-662.  doi: 10.1214/EJP.v14-634.

[11]

S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.  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. doi: 10.1017/CBO9780511750489.

[13]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  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).

[15]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5.

[16]

C. Villani, Optimal Transport: Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[17]

R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, USA, 2001, URL https://books.google.co.uk/books?id=4cI5136OdoMC.

[1]

Lingyan Cheng, Ruinan Li, Liming Wu. Exponential convergence in the Wasserstein metric $ W_1 $ for one dimensional diffusions. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5131-5148. doi: 10.3934/dcds.2020222

[2]

Liu Liu, Justyna Jarczyk, Witold Jarczyk, Weinian Zhang. Iterative roots of type $ \mathcal {T}_2 $. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022082

[3]

Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $ L_1 $ Monge-Kantorovich problem. Inverse Problems and Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037

[4]

Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058

[5]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

[6]

Anton Arnold, Beatrice Signorello. Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022009

[7]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, 2021, 29 (3) : 2445-2456. doi: 10.3934/era.2020123

[8]

Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109

[9]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017

[10]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic and Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016

[11]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic and Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485

[12]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[13]

Debora Amadori, Fatima Al-Zahrà Aqel. On the decay in $ W^{1,\infty} $ for the 1D semilinear damped wave equation on a bounded domain. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5359-5396. doi: 10.3934/dcds.2021080

[14]

Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Adrian Korban, Serap Şahinkaya, Deniz Ustun. Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021056

[15]

Chengxiang Wang, Li Zeng, Wei Yu, Liwei Xu. Existence and convergence analysis of $\ell_{0}$ and $\ell_{2}$ regularizations for limited-angle CT reconstruction. Inverse Problems and Imaging, 2018, 12 (3) : 545-572. doi: 10.3934/ipi.2018024

[16]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011

[17]

Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120

[18]

Benjamin Söllner, Oliver Junge. A convergent Lagrangian discretization for $ p $-Wasserstein and flux-limited diffusion equations. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4227-4256. doi: 10.3934/cpaa.2020190

[19]

Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control and Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001

[20]

Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (293)
  • HTML views (126)
  • Cited by (0)

Other articles
by authors

[Back to Top]