Annales Henri Poincaré

, Volume 18, Issue 2, pp 707–755 | Cite as

Small Mass Limit of a Langevin Equation on a Manifold

  • Jeremiah Birrell
  • Scott Hottovy
  • Giovanni Volpe
  • Jan Wehr


We study damped geodesic motion of a particle of mass m on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as \({m \to 0}\), its solutions converge to solutions of a limiting equation which includes a noise-induced drift term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems.


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  1. 1.
    Karatzas I., Shreve S.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. Springer, New York (2014)Google Scholar
  2. 2.
    Nelson E.: Dynamical Theories of Brownian Motion, Mathematical Notes. Princeton University Press, Princeton (1967)zbMATHGoogle Scholar
  3. 3.
    Casuso I., Khao J., Chami M., Paul-Gilloteaux P., Husain M., Duneau J.-P., Stahlberg H., Sturgis J.N., Scheuring S.: Characterization of the motion of membrane proteins using high-speed atomic force microscopy. Nat. Nanotechnol. 7(8), 525–529 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    Kärger J., Ruthven D., Theodorou D.: Diffusion in Nanoporous Materials. Wiley, New York (2012)CrossRefGoogle Scholar
  5. 5.
    Barkai E., Garini Y., Metzler R.: Strange kinetics of single molecules in living cells. Phys. Today 65(8), 29 (2012)CrossRefGoogle Scholar
  6. 6.
    Manzo C., Torreno-Pina J.A., Massignan P., Lapeyre G.J., Lewenstein M., Garcia Parajo M.F.: Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity. Phys. Rev. X. 5, 011021 (2015)Google Scholar
  7. 7.
    Ramaswamy S.: The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys. 1, 323–345 (2010)ADSCrossRefGoogle Scholar
  8. 8.
    Hsu E.: Stochastic Analysis on Manifolds, Contemporary Mathematics. American Mathematical Society, Providence (2002)Google Scholar
  9. 9.
    Stroock, D.W.: An Introduction to the Analysis of Paths on a Riemannian Manifold No. 74. American Mathematical Society, Providence (2005)Google Scholar
  10. 10.
    Van Kampen N.: Brownian motion on a manifold. J. Stat. Phys. 44(1–2), 1–24 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II: Nonequilibrium Statistical Mechanics. Springer Series in Solid-State Sciences. Springer, Berlin (2012)Google Scholar
  12. 12.
    Polettini M.: Generally covariant state-dependent diffusion. J. Stat. Mech. Theory Exp. 2013(07), P07005 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hottovy S., McDaniel A., Volpe G., Wehr J.: The smoluchowski-kramers limit of stochastic differential equations with arbitrary state-dependent friction. Commun. Math. Phys. 336(3), 1259–1283 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Smoluchowski M.V.: Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen. Z. Phys. 17, 557–585 (1916)ADSGoogle Scholar
  15. 15.
    Pavliotis G.A., Stuart A.M.: White noise limits for inertial particles in a random field. Multiscale Model. Simul. 1(4), 527–553 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chevalier C., Debbasch F.: Relativistic diffusions: a unifying approach. J. Math. Phys. 49(4), 383 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bailleul I.: A stochastic approach to relativistic diffusions. Annales de l’institut Henri Poincaré (B) 46, 760–795 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pinsky M.A.: Isotropic transport process on a Riemannian manifold. Trans. Am. Math. Soc. 218, 353–360 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pinsky M.A.: Homogenization in stochastic differential geometry. Publ. Res. Inst. Math. Sci. 17(1), 235–244 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    JØrgensen E.: Construction of the brownian motion and the ornstein-uhlenbeck process in a riemannian manifold on basis of the gangolli-mc.kean injection scheme. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 44(1), 71–87 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dowell, R.M.: Differentiable approximations to Brownian motion on manifolds, Ph.D. thesis. University of Warwick (1980)Google Scholar
  22. 22.
    Li X.-M.: Random perturbation to the geodesic equation. Ann. Probab. 44(1), 544–566 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Angst, J., Bailleul, I., Tardif, C.: Kinetic brownian motion on Riemannian manifolds. arXiv:1501.03679, (2015, arXiv preprint)
  24. 24.
    Bismut J.-M.: The hypoelliptic laplacian on the cotangent bundle. J. Am. Math. Soc. 18(2), 379–476 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Bismut J.-M.: Hypoelliptic laplacian and probability. J. Math. Soc. Japan 67(4), 1317–1357 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Herzog, D.P., Hottovy, S., Volpe, G.: The small-mass limit for Langevin dynamics with unbounded coefficients and positive friction. arXiv:1510.04187 (2015, arXiv preprint)
  27. 27.
    Seifert U.: Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75(12), 126001 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    Kobayashi S., Nomizu K.: Foundations of Differential Geometry Set. Wiley Classics Library, Wiley, New York (2009)zbMATHGoogle Scholar
  29. 29.
    Teschl, G.: Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics. American Mathematical Society, Providence (2012)Google Scholar
  30. 30.
    Kedem G.: A posteriori error bounds for two-point boundary value problems. SIAM J. Numer. Anal. 18(3), 431–448 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Freidlin M.: Some remarks on the Smoluchowski–Kramers approximation. J. Stat. Phys. 117(3), 617–634 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ortega, J.: Matrix Theory: A Second Course. University Series in Mathematics. Springer US (2013)Google Scholar
  33. 33.
    Wilcox R.M.: Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8(4), 962 (1967)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Volpe G., Wehr J.: Effective drifts in dynamical systems with multiplicative noise: A review of recent progress. Rep. Prog. Phys. 79(5), 053901 (2016)ADSCrossRefGoogle Scholar
  35. 35.
    Lee, J.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Springer, New York (2013)Google Scholar
  36. 36.
    Lee, J.: Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics. Springer, New York (2006)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Jeremiah Birrell
    • 1
  • Scott Hottovy
    • 2
  • Giovanni Volpe
    • 3
    • 4
  • Jan Wehr
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.Soft Matter Lab, Department of PhysicsBilkent UniversityAnkaraTurkey
  4. 4.UNAM, National Nanotechnology Research CenterBilkent UniversityAnkaraTurkey

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