# American Institute of Mathematical Sciences

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 & 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]. Google Scholar [2] K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd edition, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar [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.  Google Scholar [4] P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014, With an introduction to regularity structures.  Google Scholar [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.  Google Scholar [6] B. Helffer, Spectral Theory and Its Applications, vol. 139 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2013.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. , 202 (2009), ⅳ+141pp.  Google Scholar [15] C. Villani, Optimal Transport, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, Old and new.  Google Scholar [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.  Google Scholar

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##### 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]. Google Scholar [2] K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, 2nd edition, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar [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.  Google Scholar [4] P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014, With an introduction to regularity structures.  Google Scholar [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.  Google Scholar [6] B. Helffer, Spectral Theory and Its Applications, vol. 139 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2013.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. , 202 (2009), ⅳ+141pp.  Google Scholar [15] C. Villani, Optimal Transport, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, Old and new.  Google Scholar [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.  Google Scholar
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