Open Access

# Hilbert space hypocoercivity for the Langevin dynamics revisited

### Abstract

We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics via studying the longtime behavior of the strongly continuous contraction semigroup solving the associated Kolmogorov (backward) equation as an abstract Cauchy problem. This hypocoercivity result is proven in previous works before by Dolbeault, Mouhot and Schmeiser in the corresponding dual Fokker-Planck framework, but without including domain issues of the appearing operators. In our elaboration, we include the domain issues and additionally compute the rate of convergence in dependence of the damping coefficient. Important statements for the complete elaboration are the m-dissipativity results for the Langevin operator established by Conrad and the first named author of this article as well as the essential selfadjointness results for generalized Schrödinger operators by Wielens or Bogachev, Krylov and Röckner. We emphasize that the chosen Kolmogorov approach is natural. Indeed, techniques from the theory of (generalized) Dirichlet forms imply a stochastic representation of the Langevin semigroup as the transition kernel of diffusion process which provides a martingale solution to the Langevin equation. Hence an interesting connection between the theory of hypocoercivity and the theory of (generalized) Dirichlet forms is established besides.

Key words: Hypocoercivity, exponential rate of convergence, Langevin dynamics, Kolmogorov equation, operator semigroups, generalized Dirichlet forms, hypoellipticity, Poincar'e inequality, Fokker-Planck equation.

### Article Information

 Title Hilbert space hypocoercivity for the Langevin dynamics revisited Source Methods Funct. Anal. Topology, Vol. 22 (2016), no. 2, 152-168 MathSciNet MR3522857 zbMATH 06665385 Milestones Received 25/09/2015 Copyright The Author(s) 2016 (CC BY-SA)

### Authors Information

Martin Grothaus
Mathematics Department, University of Kaiserslautern, P.O.Box 3049, 67653 Kaiserslautern, Germany

Patrik Stilgenbauer
Mathematics Department, University of Kaiserslautern, P.O.Box 3049, 67653 Kaiserslautern, Germany

### Citation Example

Martin Grothaus and Patrik Stilgenbauer, Hilbert space hypocoercivity for the Langevin dynamics revisited, Methods Funct. Anal. Topology 22 (2016), no. 2, 152-168.

### BibTex

@article {MFAT847,
AUTHOR = {Grothaus, Martin and Stilgenbauer, Patrik},
TITLE = {Hilbert space hypocoercivity for the Langevin dynamics revisited},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {22},
YEAR = {2016},
NUMBER = {2},
PAGES = {152-168},
ISSN = {1029-3531},
MRNUMBER = {MR3522857},
ZBLNUMBER = {06665385},
URL = {http://mfat.imath.kiev.ua/article/?id=847},
}

### References

1. Hans Wilhelm Alt, Lineare Funktionalanalysis, Springer-Verlag Berlin Heidelberg, 2006. CrossRef
2. F. Baudoin, Bakry-Emery meet Villani, 2013.  arXiv:1308.4938
3. Dominique Bakry, Franck Barthe, Patrick Cattiaux, and Arnaud Guillin, A simple proof of the Poincare inequality for a large class of probability measures including the log-concave case, Electron. Commun. Probab. 13 (2008), 60-66.  MathSciNet CrossRef
4. Dominique Bakry, Patrick Cattiaux, and Arnaud Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincare, J. Funct. Anal. 254 (2008), no. 3, 727-759.  MathSciNet CrossRef
5. William Beckner, A generalized Poincare inequality for Gaussian measures, Proc. Amer. Math. Soc. 105 (1989), no. 2, 397-400.  MathSciNet CrossRef
6. Vladimir I. Bogachev, Nicolai V. Krylov, and Michael Rockner, Elliptic regularity and essential self-adjointness of Dirichlet operators on $R^ n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 3, 451-461.  MathSciNet
7. Florian Conrad and Martin Grothaus, Construction of $N$-particle Langevin dynamics for $H^ 1,\infty$-potentials via generalized Dirichlet forms, Potential Anal. 28 (2008), no. 3, 261-282.  MathSciNet CrossRef
8. Florian Conrad and Martin Grothaus, Construction, ergodicity and rate of convergence of $N$-particle Langevin dynamics with singular potentials, J. Evol. Equ. 10 (2010), no. 3, 623-662.  MathSciNet CrossRef
9. W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, The Langevin equation, World Scientific Series in Contemporary Chemical Physics, vol. 14, World Scientific Publishing Co., Inc., River Edge, NJ, 2004.  MathSciNet
10. F. Conrad, Construction and Analysis of Langevin Dynamics in Continuous Particle Systems, PhD Thesis, University of Kaiserslautern, Published by Verlag Dr. Hut, Munchen, 2011.
11. Jean Dolbeault, Axel Klar, Clement Mouhot, and Christian Schmeiser, Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes, Appl. Math. Res. Express. AMRX (2013), no. 2, 165-175.  MathSciNet
12. Jean Dolbeault, Clement Mouhot, and Christian Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris 347 (2009), no. 9-10, 511-516.  MathSciNet CrossRef
13. Jean Dolbeault, Clement Mouhot, and Christian Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc. 367 (2015), no. 6, 3807-3828.  MathSciNet CrossRef
14. Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994.  MathSciNet CrossRef
15. Masatoshi Fukushima, Dirichlet forms and Markov processes, North-Holland Mathematical Library, vol. 23, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980.  MathSciNet
16. Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985.  MathSciNet
17. Martin Grothaus and Patrik Stilgenbauer, Geometric Langevin equations on submanifolds and applications to the stochastic melt-spinning process of nonwovens and biology, Stoch. Dyn. 13 (2013), no. 4, 1350001, 34 pp.  MathSciNet CrossRef
18. Martin Grothaus and Patrik Stilgenbauer, Hypocoercivity for Kolmogorov backward evolution equations and applications, J. Funct. Anal. 267 (2014), no. 10, 3515-3556.  MathSciNet CrossRef
19. Martin Grothaus and Patrik Stilgenbauer, A hypocoercivity related ergodicity method for singularly distorted non-symmetric diffusions, Integral Equations Operator Theory 83 (2015), no. 3, 331-379.  MathSciNet CrossRef
20. Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001.  MathSciNet CrossRef
21. Frederic Herau, Short and long time behavior of the Fokker-Planck equation in a confining potential and applications, J. Funct. Anal. 244 (2007), no. 1, 95-118.  MathSciNet CrossRef
22. Frederic Herau and Francis Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151-218.  MathSciNet CrossRef
23. Bernard Helffer and Francis Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005.  MathSciNet
24. Tony Lelievre, Mathias Rousset, and Gabriel Stoltz, Free energy computations, Imperial College Press, London, 2010.  MathSciNet CrossRef
25. Tony Lelievre, Mathias Rousset, and Gabriel Stoltz, Langevin dynamics with constraints and computation of free energy differences, Math. Comp. 81 (2012), no. 280, 2071-2125.  MathSciNet CrossRef
26. Pierre Monmarche, Hypocoercive relaxation to equilibrium for some kinetic models, Kinet. Relat. Models 7 (2014), no. 2, 341-360.  MathSciNet CrossRef
27. Zhi Ming Ma and Michael Rockner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992.  MathSciNet CrossRef
28. J. C. Mattingly and A. M. Stuart, Geometric ergodicity of some hypo-elliptic diffusions for particle motions, Markov Process. Related Fields 8 (2002), no. 2, 199-214.  MathSciNet
29. P. A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis, Mat. Contemp. 19 (2000), 1-29.  MathSciNet
30. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.  MathSciNet CrossRef
31. Gert K. Pedersen, Analysis now, Graduate Texts in Mathematics, vol. 118, Springer-Verlag, New York, 1989.  MathSciNet CrossRef
32. Michael Rockner, $L^ p$-analysis of finite and infinite-dimensional diffusion operators, Stochastic PDEs and Kolmogorov equations in infinite dimensions (Cetraro, 1998), Lecture Notes in Math., vol. 1715, Springer, Berlin, 1999, pp. 65-116.  MathSciNet CrossRef
33. Michael Rockner and Feng-Yu Wang, Weak Poincare inequalities and $L^ 2$-convergence rates of Markov semigroups, J. Funct. Anal. 185 (2001), no. 2, 564-603.  MathSciNet CrossRef
34. Franz Schwabl, Statistical mechanics, Springer-Verlag, Berlin, 2006.  MathSciNet
35. Wilhelm Stannat, The theory of generalized Dirichlet forms and its applications in analysis and stochastics, Mem. Amer. Math. Soc. 142 (1999), no. 678, viii+101.  MathSciNet CrossRef
36. P. Stilgenbauer, The Stochastic Analysis of Fiber Lay-Down Models: An Interplay between Pure and Applied Mathematics Involving Langevin Processes on Manifolds, Ergodicity for Degenerate Kolmogorov Equations and Hypocoercivity, PhD thesis, University of Kaiserslautern, Published by Verlag Dr. Hut, Munchen, 2014.
37. Gerald Trutnau, Stochastic calculus of generalized Dirichlet forms and applications to stochastic differential equations in infinite dimensions, Osaka J. Math. 37 (2000), no. 2, 315-343.  MathSciNet
38. Cedric Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141.  MathSciNet CrossRef
39. Feng-Yu Wang, Existence of the spectral gap for elliptic operators, Ark. Mat. 37 (1999), no. 2, 395-407.  MathSciNet CrossRef
40. Norbert Wielens, The essential selfadjointness of generalized Schrodinger operators, J. Funct. Anal. 61 (1985), no. 1, 98-115.  MathSciNet CrossRef
41. Liming Wu, Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems, Stochastic Process. Appl. 91 (2001), no. 2, 205-238.  MathSciNet CrossRef