April  2021, 14(2): 199-209. doi: 10.3934/krm.2021002

Captivity of the solution to the granular media equation

Univ Lyon, Université Jean Monnet, CNRS UMR 5208, Institut Camille Jordan, Maison de l'Université, 10 rue Tréfilerie, CS 82301, 42023 Saint-Átienne Cedex 2, France

* Corresponding author: Julian Tugaut

Received  June 2020 Revised  October 2020 Published  April 2021 Early access  December 2020

The goal of the current paper is to provide assumptions under which the limiting probability of the granular media equation is known when there are several stable states. Indeed, it has been proved in our previous works [17,18] that there is convergence. However, very few is known about the limiting probability, even with a small diffusion coefficient.

Citation: Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic and Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002
References:
[1]

D. BenedettoE. CagliotiJ. A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media, J. Statist. Phys., 91 (1998), 979-990.  doi: 10.1023/A:1023032000560.

[2]

S. BenachourB. RoynetteD. Talay and P. Vallois, Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos, Stochastic Process. Appl., 75 (1998), 173-201.  doi: 10.1016/S0304-4149(98)00018-0.

[3]

S. BenachourB. Roynette and P. Vallois, Nonlinear self-stabilizing processes. II. Convergence to invariant probability, Stochastic Process. Appl., 75 (1998), 203-224.  doi: 10.1016/S0304-4149(98)00019-2.

[4]

P. CattiauxA. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case, Probab. Theory Related Fields, 140 (2008), 19-40.  doi: 10.1007/s00440-007-0056-3.

[5]

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

[6]

J. A. CarrilloR. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[7]

Samuel HerrmannPeter Imkeller and Dierk Peithmann, Large deviations and a Kramers' type law for self-stabilizing diffusions, Ann. Appl. Probab., 18 (2008), 1379-1423.  doi: 10.1214/07-AAP489.

[8]

S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., 120 (2010), 1215-1246.  doi: 10.1016/j.spa.2010.03.009.

[9]

S. Herrmann and J. Tugaut, Stationary measures for self-stabilizing processes: Asymptotic analysis in the small noise limit, Electron. J. Probab., 15 (2010), 2087-2116.  doi: 10.1214/EJP.v15-842.

[10]

S. Herrmann and J. Tugaut, Self-stabilizing processes: Uniqueness problem for stationary measures and convergence rate in the small noise limit, ESAIM Probability and statistics, 16 (2012), 277-305.  doi: 10.1051/ps/2011152.

[11]

M. Kac, Probability and Related Topics in Physical Sciences, With special lectures by G. E. Uhlenbeck, A. R. Hibbs and B. van der Pol. Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, Colo., 1957, Vol. I Interscience Publishers, London-New York 1959.

[12]

F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process. Appl., 95 (2001), 109-132.  doi: 10.1016/S0304-4149(01)00095-3.

[13]

Fl orent Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes, Ann. Appl. Probab., 13 (2003), 540-560.  doi: 10.1214/aoap/1050689593.

[14]

H. P. McKean. Jr, A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.

[15]

H. P. McKean. Jr, Propagation of chaos for a class of nonlinear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, Va., (1967), 41–57.

[16]

Yo zo Tamura, On asymptotic behaviors of the solution of a nonlinear diffusion equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31 (1984), 195-221. 

[17]

J. Tugaut, Convergence to the equilibria for self-stabilizing processes in double-well landscape, Ann. Probab., 41 (2013), 1427-1460.  doi: 10.1214/12-AOP749.

[18]

J. Tugaut, Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Convergence, Stochastic Processes and Their Applications, 123 (2013), 1780-1801.  doi: 10.1016/j.spa.2012.12.003.

[19]

J. Tugaut, Phase transitions of McKean-Vlasov processes in double-wells landscape, Stochastics, 86 (2014), 257-284.  doi: 10.1080/17442508.2013.775287.

[20]

J. Tugaut, Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Invariant probabilities, J. Theoret. Probab., 27 (2014), 57-79.  doi: 10.1007/s10959-012-0435-2.

[21]

J. Tugaut, Exit-problem of McKean-Vlasov diffusions in double-well landscape, J. Theoret. Probab., 31 (2018), 1013-1023.  doi: 10.1007/s10959-016-0737-x.

show all references

References:
[1]

D. BenedettoE. CagliotiJ. A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media, J. Statist. Phys., 91 (1998), 979-990.  doi: 10.1023/A:1023032000560.

[2]

S. BenachourB. RoynetteD. Talay and P. Vallois, Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos, Stochastic Process. Appl., 75 (1998), 173-201.  doi: 10.1016/S0304-4149(98)00018-0.

[3]

S. BenachourB. Roynette and P. Vallois, Nonlinear self-stabilizing processes. II. Convergence to invariant probability, Stochastic Process. Appl., 75 (1998), 203-224.  doi: 10.1016/S0304-4149(98)00019-2.

[4]

P. CattiauxA. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case, Probab. Theory Related Fields, 140 (2008), 19-40.  doi: 10.1007/s00440-007-0056-3.

[5]

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

[6]

J. A. CarrilloR. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[7]

Samuel HerrmannPeter Imkeller and Dierk Peithmann, Large deviations and a Kramers' type law for self-stabilizing diffusions, Ann. Appl. Probab., 18 (2008), 1379-1423.  doi: 10.1214/07-AAP489.

[8]

S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., 120 (2010), 1215-1246.  doi: 10.1016/j.spa.2010.03.009.

[9]

S. Herrmann and J. Tugaut, Stationary measures for self-stabilizing processes: Asymptotic analysis in the small noise limit, Electron. J. Probab., 15 (2010), 2087-2116.  doi: 10.1214/EJP.v15-842.

[10]

S. Herrmann and J. Tugaut, Self-stabilizing processes: Uniqueness problem for stationary measures and convergence rate in the small noise limit, ESAIM Probability and statistics, 16 (2012), 277-305.  doi: 10.1051/ps/2011152.

[11]

M. Kac, Probability and Related Topics in Physical Sciences, With special lectures by G. E. Uhlenbeck, A. R. Hibbs and B. van der Pol. Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, Colo., 1957, Vol. I Interscience Publishers, London-New York 1959.

[12]

F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process. Appl., 95 (2001), 109-132.  doi: 10.1016/S0304-4149(01)00095-3.

[13]

Fl orent Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes, Ann. Appl. Probab., 13 (2003), 540-560.  doi: 10.1214/aoap/1050689593.

[14]

H. P. McKean. Jr, A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.

[15]

H. P. McKean. Jr, Propagation of chaos for a class of nonlinear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, Va., (1967), 41–57.

[16]

Yo zo Tamura, On asymptotic behaviors of the solution of a nonlinear diffusion equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31 (1984), 195-221. 

[17]

J. Tugaut, Convergence to the equilibria for self-stabilizing processes in double-well landscape, Ann. Probab., 41 (2013), 1427-1460.  doi: 10.1214/12-AOP749.

[18]

J. Tugaut, Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Convergence, Stochastic Processes and Their Applications, 123 (2013), 1780-1801.  doi: 10.1016/j.spa.2012.12.003.

[19]

J. Tugaut, Phase transitions of McKean-Vlasov processes in double-wells landscape, Stochastics, 86 (2014), 257-284.  doi: 10.1080/17442508.2013.775287.

[20]

J. Tugaut, Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Invariant probabilities, J. Theoret. Probab., 27 (2014), 57-79.  doi: 10.1007/s10959-012-0435-2.

[21]

J. Tugaut, Exit-problem of McKean-Vlasov diffusions in double-well landscape, J. Theoret. Probab., 31 (2018), 1013-1023.  doi: 10.1007/s10959-016-0737-x.

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