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International Conference in Honor of the 75th Birthday of S.R.S. Varadhan

VAR75 2016: Probability and Analysis in Interacting Physical Systems pp 17–48Cite as

Multiscale Systems, Homogenization, and Rough Paths

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 283))

Abstract

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479–520], taking into account recent progress in p-variation and càdlàg rough path theory.

Dedicated to Professor S.R.S Varadhan on the occasion of his 75th birthday

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Notes

  1. 1.

    Since our limit processes here—a Brownian motion—is continuous, there is no need to work with the Skorokhod topology on D.

  2. 2.

    In view of the genuine non-linearity of rough path spaces, we refrain from writing \(\Vert \mathbf {X}- \tilde{\mathbf {X}} \Vert _{p\text {-var},[0,1]}\).

  3. 3.

    In coordinates, when \(\mathcal {B}= \mathbb {R}^m\), we have \(DV (Y_s) V (Y_s) \mathbb {X}_{s,t} = \partial _\alpha V_\gamma (Y_s) V^\alpha _\beta (Y_s) \mathbb {X}_{s,t}^{\beta ,\gamma }\) with summation over \(\alpha = 1, \ldots , d\) and \(\beta , \gamma = 1, \ldots , m.\).

  4. 4.

    Often \(B^n\) has continuous BV sample paths. Every such process is (trivially) a semimartingale (under its own filtration); the Stratonovich SDE interpretation is the one consistent with the ODE interpretation, in the sense of a Riemann-Stieltjes integral equation.

  5. 5.

    Again it suffices to work with the uniform topology on both \({\pmb {\mathscr {C}}}\) and \({\pmb {\mathscr {D}}}\).

References

  1. Alves, J.F., Freitas, J.M., Luzzatto, S., Vaienti, S.: From rates of mixing to recurrence times via large deviations. Adv. Math. 228(2), 1203–1236 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alves, J.F., Luzzatto, S., Pinheiro, V.: Markov structures and decay of correlations for non-uniformly expanding dynamical systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(6), 817–839 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves, J.F., Pinheiro, V.: Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction. Adv. Math. 223(5), 1706–1730 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anosov, D.V.: Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90, 209 (1967)

    MathSciNet  Google Scholar 

  5. Araujo, V., Melbourne, I.: Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation. ArXiv e-prints, November 2017

    Google Scholar 

  6. Araújo, V., Melbourne, I., Varandas, P.: Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps. Comm. Math. Phys. 340(3), 901–938 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bailleul, I., Catellier, R.: Rough flows and homogenization in stochastic turbulence. J. Diff. Equ. 263(8), 4894–4928 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benedicks, M., Young, L.-S.: Markov extensions and decay of correlations for certain Hénon maps. Astérisque (261):xi, 13–56, 2000. Géométrie complexe et systèmes dynamiques (Orsay, 1995)

    Google Scholar 

  9. Billingsley, P.: The Lindeberg-Lévy theorem for martingales. Proc. Amer. Math. Soc. 12, 788–792 (1961)

    MathSciNet  MATH  Google Scholar 

  10. Billingsley, P.: Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York (1999)

    Book  MATH  Google Scholar 

  11. Birkhoff, G.D.: Proof of the ergodic theorem. Proc. Natl. Acad. Sci. U.S.A. 17(12), 656–660 (1931)

    Article  MATH  Google Scholar 

  12. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, Berlin-New York (1975)

    Google Scholar 

  13. Breuillard, E., Friz, P., Huesmann, M.: From random walks to rough paths. Proc. Amer. Math. Soc. 137(10), 3487–3496 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brown, B.M.: Martingale central limit theorems. Ann. Math. Statist. 42, 59–66 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  15. Buzzi, J., Maume-Deschamps, V.: Decay of correlations for piecewise invertible maps in higher dimensions. Israel J. Math. 131, 203–220 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chernov, N., Young. L.S.: Decay of correlations for Lorentz gases and hard balls. In: Hard Ball Systems and the Lorentz Gas, volume 101 of Encyclopaedia Math. Sci., 89–120. Springer, Berlin (2000)

    Google Scholar 

  17. Chevyrev, I.: Random walks and Lévy processes as rough paths. Probab. Theor. Relat. Fields 170(3–4), 891–932 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chevyrev, I., Friz, P.K.: Canonical RDEs and general semimartingales as rough paths. Ann. Probab. 47(1), 420–463 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chevyrev, I., Friz, P.K., Korepanov, A., Melbourne, I., Zhang, H.: Deterministic homogenization for discrete time fast-slow systems under optimal moment assumptions. In preparation

    Google Scholar 

  20. Cuny, C., Merlevède, F.: Strong invariance principles with rate for “reverse” martingale differences and applications. J. Theoret. Probab. 28(1), 137–183 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Davie, A.M.: Differential equations driven by rough paths: an approach via discrete approximation. Appl. Math. Res. Express. AMRX, no. 2:Art. ID abm009, 40 p. (2007)

    Google Scholar 

  22. Denker, M., Philipp, W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Theor. Dynam. Syst. 4(4), 541–552 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Donsker,M.D.: An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc., No. 6:12 (1951)

    Google Scholar 

  24. Eagleson, G.K.: Some simple conditions for limit theorems to be mixing. Teor. Verojatnost. i Primenen. 21(3), 653–660 (1976)

    MathSciNet  MATH  Google Scholar 

  25. Friz, P.K., Gassiat, P., Lyons, T.J.: Physical Brownian motion in a magnetic field as a rough path. Trans. Amer. Math. Soc. 367(11), 7939–7955 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Friz, P.K., Hairer, M.: A Course on Rough Path Analysis, with an Introduction to Regularity Structures, Springer 2014. Universitext. Springer (2014)

    Google Scholar 

  27. Friz, P.K., Shekhar, A.: General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula. Ann. Probab. 45(4), 2707–2765 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Friz, P.K., Victoir, N.: A variation embedding theorem and applications. J. Funct. Anal. 239(2), 631–637 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Friz, P.K., Victoir, N.: Multidimensional Stochastic Processes as Rough Paths. Cambridge Studies in Advanced Mathematics, vol. 120. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  30. Friz, P.K., Zhang, H.: Differential equations driven by rough paths with jumps. J. Differ. Equ. 264(10), 6226–6301 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gordin, M.I.: The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739–741 (1969)

    MathSciNet  MATH  Google Scholar 

  32. Gottwald, G.A., Melbourne, I.: Homogenization for deterministic maps and multiplicative noise. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469(2156), 20130201 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gottwald, G.A., Melbourne, I.: Central limit theorems and suppression of anomalous diffusion for systems with symmetry. Nonlinearity 29(10), 2941–2960 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gouëzel, S.: Central limit theorem and stable laws for intermittent maps. Probab. Theor. Relat. Fields 128(1), 82–122 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gouëzel, S.: Statistical properties of a skew product with a curve of neutral points. Ergod. Theor. Dynam. Syst. 27(1), 123–151 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180(1), 119–140 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  37. Jakubowski, A., Mémin, J., Pagès, G.: Convergence en loi des suites d’intégrales stochastiques sur l’espace \({ D}^1\) de Skorokhod. Probab. Theor. Relat. Fields 81(1), 111–137 (1989)

    MATH  Google Scholar 

  38. Keller, G.: Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrsch. Verw. Gebiete 69(3), 461–478 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kelly, D.: Rough path recursions and diffusion approximations. Ann. Appl. Probab. 26(1), 424–461 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kelly, D., Melbourne, I.: Smooth approximation of stochastic differential equations. Ann. Probab. 44(1), 479–520 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kelly, D., Melbourne, I.: Deterministic homogenization for fast-slow systems with chaotic noise. J. Funct. Anal. 272(10), 4063–4102 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104(1), 1–19 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  43. Korepanov, A., Kosloff, Z., Melbourne, I.: Martingale-coboundary decomposition for families of dynamical systems. Annales l’Institut H Poincare. Anal. Non Lineaire 35(1), 859–885 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Korepanov, A., Kosloff, Z., Melbourne, I.: Deterministic homogenization for families of fast-slow systems. In preparation

    Google Scholar 

  45. Krzyżewski, K., Szlenk, W.: On invariant measures for expanding differentiable mappings. Studia Math. 33, 83–92 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kurtz, T.G., Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19(3), 1035–1070 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  47. Liverani, C.: Central limit theorem for deterministic systems. In: International Conference on Dynamical Systems (Montevideo, 1995), volume 362 of Pitman Res. Notes Math. Ser., pp. 56–75. Longman, Harlow (1996)

    Google Scholar 

  48. Liverani, C., Saussol, B., Vaienti, S.: A probabilistic approach to intermittency. Ergod. Theor. Dynam. Syst. 19(3), 671–685 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lyons, T.J.: Differential equations driven by rough signals. Rev. Mat. Iberoam. 14(2), 215–310 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  50. Maxwell, M., Woodroofe, M.: Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28(2), 713–724 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  51. McLeish, D.L.: Dependent central limit theorems and invariance principles. Ann. Probab. 2, 620–628 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  52. Melbourne, I.: Large and moderate deviations for slowly mixing dynamical systems. Proc. Amer. Math. Soc. 137(5), 1735–1741 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. Melbourne, I.: Superpolynomial and polynomial mixing for semiflows and flows. Nonlinearity 31(10), R268–R316 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  54. Melbourne, I.: Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260(1), 131–146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  55. Melbourne, I., Nicol, M.: Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360(12), 6661–6676 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. Melbourne, I., Stuart, A.M.: A note on diffusion limits of chaotic skew-product flows. Nonlinearity 24(4), 1361–1367 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  57. Melbourne, I., Török, A.: Statistical limit theorems for suspension flows. Israel J. Math. 144, 191–209 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  58. Melbourne, I., Török, A.: Convergence of moments for Axiom A and non-uniformly hyperbolic flows. Ergod. Theor. Dynam. Syst. 32(3), 1091–1100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  59. Melbourne, I., Varandas, P.: A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion. Stoch. Dyn. 16(3), 1660012 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  60. Melbourne, I., Zweimüller, R.: Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems. Ann. Inst. Henri Poincaré Probab. Stat. 51(2), 545–556 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  61. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque, 187–188(268) (1990)

    Google Scholar 

  62. Pavliotis, G.A., Stuart, A.M.: Multiscale methods, volume 53 of Texts in Applied Mathematics. Springer, New York, 2008. Averaging and homogenization

    Google Scholar 

  63. Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74(2), 189–197 (1980)

    Article  MathSciNet  Google Scholar 

  64. Ratner, M.: The central limit theorem for geodesic flows on \(n\)-dimensional manifolds of negative curvature. Israel J. Math. 16, 181–197 (1973)

    MathSciNet  MATH  Google Scholar 

  65. Rio, E.: Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques & Applications (Berlin) (Mathematics & Applications), vol. 31. Springer, Berlin (2000)

    Google Scholar 

  66. Ruelle, D.: Thermodynamic formalism, volume 5 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading, Mass. (1978). The mathematical structures of classical equilibrium statistical mechanics, With a foreword by Giovanni Gallavotti and Gian-Carlo Rota

    Google Scholar 

  67. Ruelle, D.: The thermodynamic formalism for expanding maps. Comm. Math. Phys. 125(2), 239–262 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  68. Rychlik, M.: Bounded variation and invariant measures. Studia Math. 76(1), 69–80 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  69. Saussol, B.: Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math. 116, 223–248 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  70. Sinaĭ, Ja.G.: Gibbs measures in ergodic theory. Uspehi Mat. Nauk. 27(4(166)), 21–64 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  71. Smale, S.: Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 747–817 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  72. Tyran-Kamińska, M.: An invariance principle for maps with polynomial decay of correlations. Comm. Math. Phys. 260(1), 1–15 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  73. Williams, D.R.E.: Path-wise solutions of stochastic differential equations driven by Lévy processes. Rev. Mat. Iberoamericana 17(2), 295–329 (2001)

    Google Scholar 

  74. Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2), 147(3), 585–650 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  75. Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  76. Young, L.-S.: What are SRB measures, and which dynamical systems have them? J. Statist. Phys. 108(5–6), 733–754 (2002). Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays

    Google Scholar 

  77. Young, L.-S.: Generalizations of SRB measures to nonautonomous, random, and infinite dimensional systems. J. Stat. Phys. 166(3–4), 494–515 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  78. Zweimüller, R.: Mixing limit theorems for ergodic transformations. J. Theoret. Probab. 20(4), 1059–1071 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I.C. is funded by a Junior Research Fellowship of St John’s College, Oxford. P.K.F. acknowledges partial support from the ERC, CoG-683164, the Einstein Foundation Berlin, and DFG research unit FOR2402. A.K. and I.M. acknowledge partial support from the European Advanced Grant StochExtHomog (ERC AdG 320977). H.Z. is supported by the Chinese National Postdoctoral Program for Innovative Talents No: BX20180075. H.Z. thanks the Institute für Mathematik, TU Berlin, for its hospitality.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom

    Ilya Chevyrev

  2. Institut für Mathematik, Technische Universität Berlin, and Weierstraß–Institut für Angewandte Analysis und Stochastik, Berlin, Germany

    Peter K. Friz

  3. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

    Alexey Korepanov & Ian Melbourne

  4. Institute of Mathematics, Fudan University, and Technische Universität Berlin, Shanghai, 200433, China

    Huilin Zhang

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  1. Ilya Chevyrev
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  2. Peter K. Friz
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  3. Alexey Korepanov
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  4. Ian Melbourne
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  5. Huilin Zhang
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Corresponding author

Correspondence to Peter K. Friz .

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Editors and Affiliations

  1. Institute of Mathematics, Technische Universität Berlin, Berlin, Germany

    Peter Friz

  2. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Wolfgang König

  3. Institute for Mathematical Stochastics, University of Münster, Münster, Germany

    Chiranjib Mukherjee

  4. CEREMADE, UMR CNRS 7534, Université Paris Dauphine - PSL, Paris, France

    Stefano Olla

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Chevyrev, I., Friz, P.K., Korepanov, A., Melbourne, I., Zhang, H. (2019). Multiscale Systems, Homogenization, and Rough Paths. In: Friz, P., König, W., Mukherjee, C., Olla, S. (eds) Probability and Analysis in Interacting Physical Systems. VAR75 2016. Springer Proceedings in Mathematics & Statistics, vol 283. Springer, Cham. https://doi.org/10.1007/978-3-030-15338-0_2

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