April  2018, 11(2): 357-395. doi: 10.3934/krm.2018017

Invariant measures for a stochastic Fokker-Planck equation

1. 

IRMAR, UMR CNRS 6625, ÉNS Rennes, Bruz, France

2. 

Institut Camille Jordan, UMR CNRS 5208, Université Lyon 1 & INRIA, Villeurbanne, France

3. 

Institut Camille Jordan, UMR CNRS 5208, CNRS & Université Lyon 1, Villeurbanne, France

* Corresponding author: L. Miguel Rodrigues

Received  March 2017 Revised  April 2017 Published  January 2018

Fund Project: Research of Sylvain De Moor was partially supported by the ANR project STOSYMAP. Research of L. Miguel Rodrigues was partially supported by the ANR project BoND ANR-13-BS01-0009-01. Research of Julien Vovelle was partially supported by the ANR projects STOSYMAP and STAB.

We study a kinetic Vlasov/Fokker-Planck equation perturbed by a stochastic forcing term. When the noise intensity is not too large, we solve the corresponding Cauchy problem in a space of functions ensuring good localization in the velocity variable. Then we show under similar conditions that the generated dynamics, with prescribed total mass, admits a unique invariant measure which is exponentially mixing. The proof relies on hypocoercive estimates and hypoelliptic regularity. At last we provide an explicit example showing that our analytic framework does require some smallness condition on the noise intensity.

Citation: 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
References:
[1]

P. Cardaliaguet, F. Delarue, J. -M. Lasry and P. -L. Lions, The master equation and the convergence problem in mean field games, arXiv: 1509.02505 [math].

[2]

K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd edition, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990.

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1992.

[4]

P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014, With an introduction to regularity structures.

[5]

T. Gallay and C. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $\bf{R^2} $, Arch. Ration. Mech. Anal., 163 (2002), 209-258.  doi: 10.1007/s002050200200.

[6]

B. Helffer, Spectral Theory and Its Applications, vol. 139 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2013.

[7]

F. Hérau and L. Thomann, On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential, J. Funct. Anal. , 271 (2016), 1301-1340, URL http://dx.doi.org/10.1016/j.jfa.2016.04.030. doi: 10.1016/j.jfa.2016.04.030.

[8]

H. J. Hwang and J. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 681-691, URL http://dx.doi.org/10.3934/dcdsb.2013.18.681. doi: 10.3934/dcdsb.2013.18.681.

[9]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998, URL http://dx.doi.org/10.1088/0951-7715/19/4/011. doi: 10.1088/0951-7715/19/4/011.

[10]

B. Øksendal, Stochastic Differential Equations, 5th edition, Universitext, Springer-Verlag, Berlin, 1998, URL http://dx.doi.org/10.1007/978-3-662-03620-4, An introduction with applications.

[11]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I, 2nd edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980, Functional analysis.

[12]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 1999.

[13]

R. L. Schilling and L. Partzsch, Brownian Motion, 2nd edition, De Gruyter Graduate, De Gruyter, Berlin, 2014, An introduction to stochastic processes, With a chapter on simulation by Björn Böttcher.

[14]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. , 202 (2009), ⅳ+141pp.

[15]

C. Villani, Optimal Transport, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, Old and new.

[16]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, The Annals of Mathematical Statistics, 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.

show all references

References:
[1]

P. Cardaliaguet, F. Delarue, J. -M. Lasry and P. -L. Lions, The master equation and the convergence problem in mean field games, arXiv: 1509.02505 [math].

[2]

K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd edition, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990.

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1992.

[4]

P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014, With an introduction to regularity structures.

[5]

T. Gallay and C. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $\bf{R^2} $, Arch. Ration. Mech. Anal., 163 (2002), 209-258.  doi: 10.1007/s002050200200.

[6]

B. Helffer, Spectral Theory and Its Applications, vol. 139 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2013.

[7]

F. Hérau and L. Thomann, On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential, J. Funct. Anal. , 271 (2016), 1301-1340, URL http://dx.doi.org/10.1016/j.jfa.2016.04.030. doi: 10.1016/j.jfa.2016.04.030.

[8]

H. J. Hwang and J. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 681-691, URL http://dx.doi.org/10.3934/dcdsb.2013.18.681. doi: 10.3934/dcdsb.2013.18.681.

[9]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998, URL http://dx.doi.org/10.1088/0951-7715/19/4/011. doi: 10.1088/0951-7715/19/4/011.

[10]

B. Øksendal, Stochastic Differential Equations, 5th edition, Universitext, Springer-Verlag, Berlin, 1998, URL http://dx.doi.org/10.1007/978-3-662-03620-4, An introduction with applications.

[11]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I, 2nd edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980, Functional analysis.

[12]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 1999.

[13]

R. L. Schilling and L. Partzsch, Brownian Motion, 2nd edition, De Gruyter Graduate, De Gruyter, Berlin, 2014, An introduction to stochastic processes, With a chapter on simulation by Björn Böttcher.

[14]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. , 202 (2009), ⅳ+141pp.

[15]

C. Villani, Optimal Transport, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, Old and new.

[16]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, The Annals of Mathematical Statistics, 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.

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