Perturbation of Conservation Laws and Averaging on Manifolds

  • Xue-Mei LiEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


We prove a stochastic averaging theorem for stochastic differential equations in which the slow and the fast variables interact. The approximate Markov fast motion is a family of Markov process with generator \({\mathscr L}_x\) for which we obtain a quantitative locally uniform law of large numbers and obtain the continuous dependence of their invariant measures on the parameter x. These results are obtained under the assumption that \({\mathscr L}_x\) satisfies Hörmander’s bracket conditions, or more generally \({\mathscr L}_x\) is a family of Fredholm operators with sub-elliptic estimates. For stochastic systems in which the slow and the fast variable are not separate, conservation laws are essential ingredients for separating the scales in singular perturbation problems we demonstrate this by a number of motivating examples, from mathematical physics and from geometry, where conservation laws taking values in non-linear spaces are used to deduce slow-fast systems of stochastic differential equations.


  1. 1.
    Angst, J., Bailleul, I, Tardif, C.: Kinetic Brownian motion on Riemannian manifolds. Electron. J. Probab. 20(110), 40 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Albeverio, S., Daletskii, A., Kalyuzhnyi, A.: Random Witten Laplacians: traces of semigroups, L 2-Betti numbers and index. J. Eur. Math. Soc. (JEMS) 10(3), 571–599 (2008)Google Scholar
  3. 3.
    Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)zbMATHGoogle Scholar
  4. 4.
    Arnaudon, M.: Semi-martingales dans les espaces homogènes. Ann. Inst. H. Poincaré Probab. Statist. 29(2), 269–288 (1993)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Atiyah, M.F.: Algebraic topology and operators in Hilbert space. In: Atiyah, M.F., Taam, C.T., et al. (eds.) Lectures in Modern Analysis and Applications. I, pp. 101–121. Springer, Berlin (1969)Google Scholar
  6. 6.
    Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348. Springer, Cham (2014)zbMATHGoogle Scholar
  7. 7.
    Bally, V., Caramellino, L.: Asymptotic development for the CLT in total variation distance. Bernoulli 22(4), 2442–2485 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Barret, F., von Renesse, M.: Averaging principle for diffusion processes via Dirichlet forms. Potential Anal. 41(4), 1033–1063 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Baudoin, F.: An Introduction to the Geometry of Stochastic Flows. Imperial College Press, London (2004)zbMATHGoogle Scholar
  10. 10.
    Baudoin, F., Hairer, M., Teichmann, J.: Ornstein-Uhlenbeck processes on Lie groups. J. Funct. Anal. 255(4), 877–890 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bérard-Bergery, L., Bourguignon, J.-P.: Laplacians and Riemannian submersions with totally geodesic fibres. Illinois J. Math. 26(2), 181–200 (1982)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Berglund, N., Gentz, B.: Noise-Induced phenomena in Slow-Fast Dynamical Systems. Probability and Its Applications (New York). Springer, London (2006). A sample-paths approachGoogle Scholar
  13. 13.
    Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, New York (1992)zbMATHGoogle Scholar
  14. 14.
    Birrell, J., Hottovy, S., Volpe, G., Wehr, J.: Small mass limit of a Langevin equation on a manifold. Ann. Henri Poincaré 18(2), 707–755 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Bismut, J.-M.: The hypoelliptic Laplacian on a compact Lie group. J. Funct. Anal. 255(9), 2190–2232 (2008)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Bismut, J.-M., Lebeau, G.: Laplacien hypoelliptique et torsion analytique. C. R. Math. Acad. Sci. Paris 341(2), 113–118 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Borodin, A.N.: A limit theorem for the solutions of differential equations with a random right-hand side. Teor. Verojatnost. i Primenen. 22(3), 498–512 (1977)MathSciNetGoogle Scholar
  18. 18.
    Borodin, A.N., Freidlin, M.I.: Fast oscillating random perturbations of dynamical systems with conservation laws. Ann. Inst. H. Poincaré Probab. Statist. 31(3), 485–525 (1995)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Cass, T., Friz, P.: Densities for rough differential equations under Hörmander’s condition. Ann. Math. (2) 171(3), 2115–2141 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Catellier, R., Gubinelli, M.: Averaging along irregular curves and regularisation of ODEs. Stochastic Process. Appl. 126(8), 2323–2366 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci flow: techniques and applications. Part III. In: Geometric-Analytic Aspects. Mathematical Surveys and Monographs, vol. 163. American Mathematical Society, Providence (2010)Google Scholar
  22. 22.
    Crisan, D., Ottobre, M.: Pointwise gradient bounds for degenerate semigroups (of UFG type). Proc. A 472(2195), 20160442, 23 (2016)MathSciNetzbMATHGoogle Scholar
  23. 23.
    David Elworthy, K., Le Jan, Y., Li, X.-M.: The Geometry of Filtering. Frontiers in Mathematics. Birkhäuser, Basel (2010)zbMATHGoogle Scholar
  24. 24.
    Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for linear kinetic equations conserving mass. Trans. Am. Math. Soc. 367(6), 3807–3828 (2015)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Dolgopyat, D., Kaloshin, V., Koralov, L.: Sample path properties of the stochastic flows. Ann. Probab. 32(1A), 1–27 (2004)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Dowell, R.M.: Differentiable approximations to Brownian motion on manifolds. PhD thesis, University of Warwick (1980)Google Scholar
  27. 27.
    Duong, M.H., Lamacz, A., Peletier, M.A., Sharma, U.: Variational approach to coarse-graining of generalized gradient flows. Calc. Var. 56, 100 (2017). Published first onlineGoogle Scholar
  28. 28.
    Eckmann, J.-P., Hairer, M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Commun. Math. Phys. 219(3), 523–565 (2001)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Elworthy, K.D., Le Jan, Y., Li, X.-M.: On the Geometry of Diffusion Operators and Stochastic Flows. Lecture Notes in Mathematics, vol. 1720. Springer, New York (1999)Google Scholar
  30. 30.
    Elworthy, K.D.: Stochastic Differential Equations on Manifolds. London Mathematical Society Lecture Note Series, vol. 70. Cambridge University Press, Cambridge (1982)Google Scholar
  31. 31.
    Émery, M.: Stochastic Calculus in Manifolds. Universitext. Springer, Berlin (1989) With an appendix by P.-A. MeyerzbMATHGoogle Scholar
  32. 32.
    Freidlin, M.I.: The averaging principle and theorems on large deviations. Uspekhi Mat. Nauk 33(5(203)), 107–160, 238 (1978)MathSciNetGoogle Scholar
  33. 33.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260. Springer, New York (1984). Translated from the Russian by Joseph SzücsGoogle Scholar
  34. 34.
    Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, 3rd edn. Springer, Heidelberg (2012). Translated from the 1979 Russian original by Joseph SzücsGoogle Scholar
  35. 35.
    Friz, P., Gassiat, P., Lyons, T.: Physical Brownian motion in a magnetic field as a rough path. Trans. Am. Math. Soc. 367(11), 7939–7955 (2015)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Fu, H., Duan, J.: An averaging principle for two-scale stochastic partial differential equations. Stoch. Dyn. 11(2–3), 353–367 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Fu, H., Liu, J.: Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations. J. Math. Anal. Appl. 384(1), 70–86 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Gonzales-Gargate, I.I., Ruffino, P.R.: An averaging principle for diffusions in foliated spaces. Ann. Probab. 44(1), 567–588 (2016)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Greene, R.E., Wu, H.: C approximations of convex, subharmonic, and plurisubharmonic functions. Ann. Sci. École Norm. Sup. (4) 12(1), 47–84 (1979)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Gu, Y., Mourrat, J.-C.: Pointwise two-scale expansion for parabolic equations with random coefficients. Probab. Theory Relat. Fields 166(1–2), 585–618 (2016)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Hairer, M., Pavliotis, G.A.: Periodic homogenization for hypoelliptic diffusions. J. Statist. Phys. 117(1–2), 261–279 (2004)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Hairer, M., Mattingly, J.C.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. (2) 164(3), 993–1032 (2006)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Hairer, M., Pardoux, E.: Homogenization of periodic linear degenerate PDEs. J. Funct. Anal. 255(9), 2462–2487 (2008)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Hairer, M., Pillai, N.S.: Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 47(2), 601–628 (2011)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Hairer, M., Mattingly, J.C., Scheutzow, M.: Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Related Fields 149(1–2), 223–259 (2011)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Has’minskii, R.Z.: On the principle of averaging the Itô’s stochastic differential equations. Kybernetika (Prague) 4, 260–279 (1968)MathSciNetGoogle Scholar
  47. 47.
    Helland, I.S.: Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist. 9(2), 79–94 (1982)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Högele, M., Ruffino, P.: Averaging along foliated Lévy diffusions. Nonlinear Anal. 112, 1–14 (2015)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Ikeda, N., Ogura, Y.: A degenerating sequence of Riemannian metrics on a manifold and their Brownian motions. In: Diffusion Processes and Related Problems in Analysis, Vol. I. Progress in probability, vol. 22, pp. 293–312. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  51. 51.
    Kelly, D., Melbourne, I.: Deterministic homogenization for fast-slow systems with chaotic noise. J. Funct. Anal. 272(10), 4063–4102 (2017)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Kifer, Y.: Random Perturbations of Dynamical Systems. Progress in Probability and Statistics, vol. 16. Birkhäuser, Boston (1988)zbMATHGoogle Scholar
  53. 53.
    Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104(1), 1–19 (1986)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Vol I. Interscience Publishers, a division of John Wiley & Sons, New York/London (1963)Google Scholar
  55. 55.
    Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 345. Springer, Heidelberg (2012). Time symmetry and martingale approximationzbMATHGoogle Scholar
  56. 56.
    Korepanov, A., Kosloff, Z., Melbourne, I.: Martingale-coboundary decomposition for families of dynamical systems. Annales I’Institut Henri Poincaré, Analyse non-linéaire, pp. 859–885 (2018)Google Scholar
  57. 57.
    Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Kuehn, C.: Multiple Time Scale Dynamics. Volume 191 of Applied Mathematical Sciences. Springer, Cham (2015)Google Scholar
  59. 59.
    Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32(1), 1–76 (1985)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Kurtz, T.G.: A general theorem on the convergence of operator semigroups. Trans. Am. Math. Soc. 148, 23–32 (1970)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Langevin, P.: Sur la thèorie du mouvement brownien. C. R. Acad. Sci. (Paris), 146 (1908)Google Scholar
  62. 62.
    Li, X.-M.: An averaging principle for a completely integrable stochastic Hamiltonian system. Nonlinearity 21(4), 803–822 (2008)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Li, X.-M.: Effective diffusions with intertwined structures. arxiv:1204.3250 (2012)Google Scholar
  64. 64.
    Li, X.-M.: Random perturbation to the geodesic equation. Ann. Probab. 44(1), 544–566 (2015)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Li, X.-M.: Homogenisation on homogeneous spaces. J. Math. Soc. Jpn. 70(2), 519–572 (2018)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Li, X.-M.: Limits of random differential equations on manifolds. Probab. Theory Relat. Fields 166(3–4), 659–712 (2016). MathSciNetzbMATHGoogle Scholar
  67. 67.
    Liverani, C., Olla, S.: Toward the Fourier law for a weakly interacting an harmonic crystal. J. Am. Math. Soc. 25(2), 555–583 (2012)zbMATHGoogle Scholar
  68. 68.
    Mazzeo, R.R., Melrose, R.B.: The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration. J. Differ. Geom. 31(1), 185–213 (1990)zbMATHGoogle Scholar
  69. 69.
    Mischler, S., Mouhot, C.: Exponential stability of slowly decaying solutions to the kinetic- Fokker-Planck equation. Arch. Ration. Mech. Anal. 221(2), 677–723 (2016)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Myers, S.B., Steenrod, N.E.: The group of isometries of a Riemannian manifold. Ann. Math. (2) 40(2), 400–416 (1939)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)zbMATHGoogle Scholar
  72. 72.
    Ogura, Y., Taniguchi, S.: A probabilistic scheme for collapse of metrics. J. Math. Kyoto Univ. 36(1), 73–92 (1996)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Papanicolaou, G.C., Kohler, W.: Asymptotic theory of mixing stochastic ordinary differential equations. Commun. Pure Appl. Math. 27, 641–668 (1974)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Papanicolaou, G.C., Stroock, D., Varadhan, S.R.S.: Martingale approach to some limit theorems. In: Papers from the Duke Turbulence Conference (Duke University,1976), ii+120pp. Duke University, Durham (1977)Google Scholar
  75. 75.
    Papanicolaou, G.C., Varadhan, S.R.S.: A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations. Commun. Pure Appl. Math. 26, 497–524 (1973)MathSciNetzbMATHGoogle Scholar
  76. 76.
    Pavliotis, G.A., Stuart, A.M.: Multiscale Methods: Averaging and Homogenization. Texts in Applied Mathematics, vol. 53. Springer, New York (2008)Google Scholar
  77. 77.
    Ruffino, P.R.: Application of an averaging principle on foliated diffusions: topology of the leaves. Electron. Commun. Probab. 20(28), 5 (2015)MathSciNetzbMATHGoogle Scholar
  78. 78.
    Schoen, R., Yau, S.-T.: Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge (1994). Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu, Translated from the Chinese by Ding and S. Y. Cheng, Preface translated from the Chinese by Kaising TsoGoogle Scholar
  79. 79.
    Skorokhod, A.V., Hoppensteadt, F.C., Salehi, H.: Random Perturbation Methods with Applications in Science and Engineering. Applied Mathematical Sciences, vol. 150. Springer, New York (2002)zbMATHGoogle Scholar
  80. 80.
    Stratonovich, R.L.: Selected problems in the theory of fluctuations in radio engineering. Sov. Radio, Moscow (1961). In RussianGoogle Scholar
  81. 81.
    Stratonovich, R.L.: Topics in the Theory of Random Noise. Vol. I: General Theory of Random Processes. Nonlinear transformations of signals and noise. Revised English edition. Translated from the Russian by Richard A. Silverman. Gordon and Breach Science Publishers, New York/London (1963)Google Scholar
  82. 82.
    Stroock, D., Varadhan, S.R.S.: Theory of diffusion processes. In: Stochastic Differential Equations. C.I.M.E. Summer Schools, vol. 77, pp. 149–191. Springer, Heidelberg (2010)Google Scholar
  83. 83.
    Tam, L.-F.: Exhaustion functions on complete manifolds. In: Lee, Y.-I., Lin, C.-S., Tsui, M.-P. (eds.) Recent Advances in Geometric Analysis. Advanced Lectures in Mathematics (ALM), vol. 11, pp. 211–215. Internat Press, Somerville (2010)Google Scholar
  84. 84.
    Tanno, S.: The first eigenvalue of the Laplacian on spheres. Tôhoku Math. J. (2) 31(2), 179–185 (1979)MathSciNetzbMATHGoogle Scholar
  85. 85.
    Uhlenbeck, G.E., Ornstein, L.S.: Brownian motion in a field of force and the diffusion model of chemical reactions. Phys. Rev. 36, 823–841 (1930)zbMATHGoogle Scholar
  86. 86.
    Urakawa, H.: The first eigenvalue of the Laplacian for a positively curved homogeneous Riemannian manifold. Compositio Math. 59(1), 57–71 (1986)MathSciNetzbMATHGoogle Scholar
  87. 87.
    van Erp, E.: The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I. Ann. Math. (2) 171(3), 1647–1681 (2010)MathSciNetzbMATHGoogle Scholar
  88. 88.
    van Erp, E.: The index of hypoelliptic operators on foliated manifolds. J. Noncommut. Geom. 5(1), 107–124 (2011)MathSciNetzbMATHGoogle Scholar
  89. 89.
    Veretennikov, A.Yu.: On an averaging principle for systems of stochastic differential equations. Mat. Sb. 181(2), 256–268 (1990)zbMATHGoogle Scholar
  90. 90.
    Villani, C.: Hypocoercive diffusion operators. In: International Congress of Mathematicians, vol. III, pp. 473–498. European Mathematical Society, Zürich (2006)Google Scholar
  91. 91.
    Weinan, E.: Principles of Multiscale Modeling. Cambridge University Press, Cambridge (2011)zbMATHGoogle Scholar
  92. 92.
    Yosida, K.: Functional Analysis. Die Grundlehren der Mathematischen Wissenschaften, Band 123. Academic Press/Springer, New York/Berlin (1965)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations