• Previous Article
    On the solvability of a semilinear higher-order elliptic problem for the vector field method in image registration
  • CPAA Home
  • This Issue
  • Next Article
    Wellposedness of a DNA replication model based on a nucleation-growth process
August  2022, 21(8): 2661-2677. doi: 10.3934/cpaa.2022066

The Łojasiewicz inequality for free energy functionals on a graph

School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

* Corresponding author

Received  August 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

Fund Project: Xiaoping Xue is supported by the Chinese Natural Science Foundation grants 11731010 and 11671109, and supported in part by NSF of China Grant No.11271099

Rencently Chow, Huang, Li and Zhou proposed discrete forms of the Fokker-Planck equations on a finite graph. As a primary step, they constructed Riemann metrics on the graph by endowing it with some kinds of weight. In this paper, we reveal the relation between these Riemann metrics and the Euclidean metric, by showing that they are locally equivalent. Moreover, various Riemann metrics have this property provided the corresponding weight satisfies a bounded condition. Based on this, we prove that the two-side Łojasiewicz inequality holds near the Gibbs distribution with Łojasiewicz exponent $ \frac{1}{2} $. Then we use it to prove the solution of the discrete Fokker-Planck equation converges to the Gibbs distribution with exponential rate. As a corollary of Łojasiewicz inequality, we show that the two-side Talagrand-type inequality holds under different Riemann metrics.

Citation: Kongzhi Li, Xiaoping Xue. The Łojasiewicz inequality for free energy functionals on a graph. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2661-2677. doi: 10.3934/cpaa.2022066
References:
[1]

J. BolteA. Daniilidis and A. Lewis, The łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim., 17 (2007), 1205-1223.  doi: 10.1137/050644641.

[2]

J. A. CarrilloR. J. McCann and C. Villani et al., Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), 971-1018.  doi: 10.4171/RMI/376.

[3]

R. CheW. HuangY. Li and P. Tetali, Convergence to global equilibrium for fokker–planck equations on a graph and talagrand-type inequalities, J. Differ. Equ., 261 (2016), 2552-2583.  doi: 10.1016/j.jde.2016.05.003.

[4]

S.-N. ChowL. DieciW. Li and H. Zhou, Entropy dissipation semi-discretization schemes for fokker–planck equations, J. Dynam. Differ. Equ., 31 (2019), 765-792.  doi: 10.1007/s10884-018-9659-x.

[5]

S.-N. ChowW. HuangY. Li and H. Zhou, Fokker–planck equations for a free energy functional or markov process on a graph, Arch. Ration. Mech. Anal., 203 (2012), 969-1008.  doi: 10.1007/s00205-011-0471-6.

[6]

S.-N. ChowW. Li and H. Zhou, Entropy dissipation of fokker-planck equations on graphs, Discret. Contin. Dynam. Syst. A, 38 (2018), 4929-4950.  doi: 10.3934/dcds.2018215.

[7]

S.-N. ChowW. Li and H. Zhou, A discrete schrödinger equation via optimal transport on graphs, J. Funct. Anal., 276 (2019), 2440-2469.  doi: 10.1016/j.jfa.2019.02.005.

[8]

J. Cui, L. Dieci and H. Zhou, Time discretizations of wasserstein-hamiltonian flows, arXiv: 2006.09187.

[9]

J. CuiS. Liu and H. Zhou, What is a stochastic hamiltonian process on finite graph? an optimal transport answer, J. Differ. Equ., 305 (2021), 428-457.  doi: 10.1016/j.jde.2021.10.009.

[10]

W. Gangbo, W. Li and C. Mou, Geodesics of minimal length in the set of probability measures on graphs, ESAIM: Control, Optim. Calc. Var., 25 (2019), 36 pp. doi: 10.1051/cocv/2018052.

[11]

C. W. Gardiner et al., Handbook of Stochastic Methods, vol. 3, Springer Berlin, 1985.

[12]

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.

[13]

Z. Li, X. Xue and D. Yu, On the łojasiewicz exponent of kuramoto model, J. Math. Phys., 56 (2015), 022704, 20 pp. doi: 10.1063/1.4908104.

[14]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les Équations aux Dérivées Partielles, (1963), 87–89.

[15]

F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Commun. Partial Differ. Equ., 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.

[16]

F. Otto and C. Villani, Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality, J. Funct. Anal., 173 (2000), 361-400.  doi: 10.1006/jfan.1999.3557.

[17]

H. Risken and J. Eberly, The fokker-planck equation, methods of solution and applications, J. Optical Soc. Amer. B Optical Physics, 2 (1985), 508. 

[18]

Z. Schuss, Singular perturbation methods in stochastic differential equations of mathematical physics, SIAM Rev., 22 (1980), 119-155.  doi: 10.1137/1022024.

[19]

A. Torgašev and M. Petrović, Lower bounds of the laplacian graph eigenvalues, Indagationes Math., 15 (2004), 589-593.  doi: 10.1016/S0019-3577(04)80021-1.

[20]

C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, Handbook Math. Fluid Dynam., 1 (2002), 71-74.  doi: 10.1016/S1874-5792(02)80004-0.

[21]

C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2003. doi: 10.1090/gsm/058.

[22]

C. Villani, Optimal Transport: Old and New, Springer Science & Business Media, 2008. doi: 10.1007/978-3-540-71050-9.

show all references

References:
[1]

J. BolteA. Daniilidis and A. Lewis, The łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim., 17 (2007), 1205-1223.  doi: 10.1137/050644641.

[2]

J. A. CarrilloR. J. McCann and C. Villani et al., Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), 971-1018.  doi: 10.4171/RMI/376.

[3]

R. CheW. HuangY. Li and P. Tetali, Convergence to global equilibrium for fokker–planck equations on a graph and talagrand-type inequalities, J. Differ. Equ., 261 (2016), 2552-2583.  doi: 10.1016/j.jde.2016.05.003.

[4]

S.-N. ChowL. DieciW. Li and H. Zhou, Entropy dissipation semi-discretization schemes for fokker–planck equations, J. Dynam. Differ. Equ., 31 (2019), 765-792.  doi: 10.1007/s10884-018-9659-x.

[5]

S.-N. ChowW. HuangY. Li and H. Zhou, Fokker–planck equations for a free energy functional or markov process on a graph, Arch. Ration. Mech. Anal., 203 (2012), 969-1008.  doi: 10.1007/s00205-011-0471-6.

[6]

S.-N. ChowW. Li and H. Zhou, Entropy dissipation of fokker-planck equations on graphs, Discret. Contin. Dynam. Syst. A, 38 (2018), 4929-4950.  doi: 10.3934/dcds.2018215.

[7]

S.-N. ChowW. Li and H. Zhou, A discrete schrödinger equation via optimal transport on graphs, J. Funct. Anal., 276 (2019), 2440-2469.  doi: 10.1016/j.jfa.2019.02.005.

[8]

J. Cui, L. Dieci and H. Zhou, Time discretizations of wasserstein-hamiltonian flows, arXiv: 2006.09187.

[9]

J. CuiS. Liu and H. Zhou, What is a stochastic hamiltonian process on finite graph? an optimal transport answer, J. Differ. Equ., 305 (2021), 428-457.  doi: 10.1016/j.jde.2021.10.009.

[10]

W. Gangbo, W. Li and C. Mou, Geodesics of minimal length in the set of probability measures on graphs, ESAIM: Control, Optim. Calc. Var., 25 (2019), 36 pp. doi: 10.1051/cocv/2018052.

[11]

C. W. Gardiner et al., Handbook of Stochastic Methods, vol. 3, Springer Berlin, 1985.

[12]

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.

[13]

Z. Li, X. Xue and D. Yu, On the łojasiewicz exponent of kuramoto model, J. Math. Phys., 56 (2015), 022704, 20 pp. doi: 10.1063/1.4908104.

[14]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les Équations aux Dérivées Partielles, (1963), 87–89.

[15]

F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Commun. Partial Differ. Equ., 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.

[16]

F. Otto and C. Villani, Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality, J. Funct. Anal., 173 (2000), 361-400.  doi: 10.1006/jfan.1999.3557.

[17]

H. Risken and J. Eberly, The fokker-planck equation, methods of solution and applications, J. Optical Soc. Amer. B Optical Physics, 2 (1985), 508. 

[18]

Z. Schuss, Singular perturbation methods in stochastic differential equations of mathematical physics, SIAM Rev., 22 (1980), 119-155.  doi: 10.1137/1022024.

[19]

A. Torgašev and M. Petrović, Lower bounds of the laplacian graph eigenvalues, Indagationes Math., 15 (2004), 589-593.  doi: 10.1016/S0019-3577(04)80021-1.

[20]

C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, Handbook Math. Fluid Dynam., 1 (2002), 71-74.  doi: 10.1016/S1874-5792(02)80004-0.

[21]

C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2003. doi: 10.1090/gsm/058.

[22]

C. Villani, Optimal Transport: Old and New, Springer Science & Business Media, 2008. doi: 10.1007/978-3-540-71050-9.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics and Games, 2021, 8 (4) : 381-402. doi: 10.3934/jdg.2021013

[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]

Alain Haraux. Some applications of the Łojasiewicz gradient inequality. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2417-2427. doi: 10.3934/cpaa.2012.11.2417

[8]

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

[9]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008

[10]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028

[11]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

[12]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028

[13]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215

[14]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009

[15]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic and Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165

[16]

Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65

[17]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079

[18]

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

[19]

Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic and Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028

[20]

Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic and Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044

2021 Impact Factor: 1.273

Article outline

[Back to Top]