Skip to main content
SpringerLink
Log in

Invariance of white noise for KdV on the line

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We consider the Korteweg–de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babin, A.V., Ilyin, A.A., Titi, E.: On the regularization mechanism for the periodic Korteweg–de Vries equation. Commun. Pure Appl. Math. 64(5), 591–648 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Brereton, J.T.: Invariant measure construction at a fixed mass. Nonlinearity 32(2), 496–558 (2019)

    MathSciNet  MATH  Google Scholar 

  3. Binder, I., Damanik, D., Goldstein, M., Lukic, M.: Almost periodicity in time of solutions of the KdV equation. Duke Math. J. 167(14), 2633–2678 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys. 166(1), 1–26 (1994)

    MATH  Google Scholar 

  5. Bourgain, J.: Invariant measures for NLS in infinite volume. Commun. Math. Phys. 210, 605–620 (2000)

    MathSciNet  MATH  Google Scholar 

  6. Brydges, D.C., Slade, G.: Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation. Commun. Math. Phys. 182(2), 485–504 (1996)

    MATH  Google Scholar 

  7. Burq, N., Thomann, L., Tzvetkov, N.: Remarks on the Gibbs measures for nonlinear dispersive equations. Ann. Fac. Sci. Toulouse Math. (6) 27(3), 527–597 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Carlen, E.A., Fröhlich, J., Lebowitz, J.: Exponential relaxation to equilibrium for a one-dimensional focusing non-linear Schrödinger equation with noise. Commun. Math. Phys. 342(1), 303–332 (2016)

    MATH  Google Scholar 

  9. Chatterjee, S.: Invariant measures and the soliton resolution conjecture. Commun. Pure Appl. Math. 67(11), 1737–1842 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Christ, M.: Nonuniqueness of weak solutions of the nonlinear Schrödinger equation. Preprint arXiv:math/0503366

  11. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company Inc, New York (1955)

    MATH  Google Scholar 

  12. Cohen, A., Kappeler, T.: Nonuniqueness for solutions of the Korteweg–de Vries equation. Trans. Am. Math. Soc. 312(2), 819–840 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Damanik, D., Goldstein, M.: On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data. J. Am. Math. Soc. 29(3), 825–856 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Deift, P.: Some open problems in random matrix theory and the theory of integrable systems. In: Integrable Systems and Random Matrices, 419–430, Contemp. Math., 458, Amer. Math. Soc., Providence, RI (2008)

  15. Deift, P.: Some open problems in random matrix theory and the theory of integrable systems. II. SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper No. 016

  16. Deng, Y.: Invariance of the Gibbs measure for the Benjamin–Ono equation. J. Eur. Math. Soc. (JEMS) 17(5), 1107–1198 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Deng, Y., Tzvetkov, N., Visciglia, N.: Invariant measures and long time behaviour for the Benjamin–Ono equation III. Commun. Math. Phys. 339(3), 815–857 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Eichinger, B., VandenBoom, T., Yuditskii, P.: KdV hierarchy via Abelian coverings and operator identities. Trans. Am. Math. Soc. Ser. B 6, 1–44 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88(2), 151–184 (1983)

    MathSciNet  MATH  Google Scholar 

  20. Gardner, C.S.: Korteweg–de Vries equation and generalizations. IV. The Korteweg–de Vries equation as a Hamiltonian system. J. Math. Phys. 12(8), 1548–1551 (1971)

    MathSciNet  MATH  Google Scholar 

  21. Genovese, G., Lucà, R., Valeri, D.: Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation. Selecta Math. (N.S.) 22(3), 1663–1702 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Hislop, P.D.: Exponential decay of two-body eigenfunctions: a review. In: Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999), pp. 265–288, Electron. J. Differ. Equ. Conf., vol. 4, Southwest Texas State Univ., San Marcos (2000)

  23. Kappeler, T., Topalov, P.: Global wellposedness of KdV in \(H^{-1}({\mathbb{T}},{\mathbb{R}})\). Duke Math. J. 135(2), 327–360 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995)

    MATH  Google Scholar 

  25. Killip, R., Visan, M.: KdV is wellposed in \(H^{-1}\). Ann. Math. (2) 190(1), 249–305 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39(240), 422–443 (1895)

    MathSciNet  MATH  Google Scholar 

  27. Klein, A.: Multiscale analysis and localization of random operators. Random Schrödinger operators, 121–159, Panor. Synthèses, 25, Soc. Math. France, Paris, (2008)

  28. Kotani, S.: Construction of KdV flow I. Tau function via Weyl function. Zh. Mat. Fiz. Anal. Geom. 14(3), 297–335 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Lebowitz, J.L., Rose, H., Speer, E.R.: Statistical mechanics of the nonlinear Schrödinger equation. J. Stat. Phys. 50(3–4), 657–687 (1988)

    MATH  Google Scholar 

  30. Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19(5), 1156–1162 (1978)

    MathSciNet  MATH  Google Scholar 

  31. Malliavin, P.: Integration and Probability. Graduate Texts in Mathematics, vol. 157. Springer, New York (1995)

    MATH  Google Scholar 

  32. McKean, H.P.: Statistical mechanics of nonlinear wave equations 3: metric transitivity for hyperbolic sine-Gordon. J. Stat. Phys. 79(3–4), 731–737 (1995)

    MathSciNet  MATH  Google Scholar 

  33. McKean, H.P.: Correction. Statistical Mechanics of Nonlinear Wave Equations (3). Metric Transitivity for Hyperbolic Sine-Gordon. J. Stat. Phys. 95(1-2), 517 (1999)

  34. McKean, H.P., Vaninsky, K.L.: Statistical mechanics of nonlinear wave equations. Trends and perspectives in applied mathematics. Appl. Math. Sci. 100, 239–264 (1994)

    MATH  Google Scholar 

  35. Minami, N.: Exponential and super-exponential localizations for one-dimensional Schrödinger operators with Lévy noise potentials. Tsukuba J. Math. 13(1), 225–282 (1989)

    MathSciNet  MATH  Google Scholar 

  36. Miura, R.M., Gardner, C.S., Kruskal, M.D.: Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys. 9(8), 1204–1209 (1968)

    MathSciNet  MATH  Google Scholar 

  37. Nahmod, A.R., Oh, T., Rey-Bellet, L., Staffilani, G.: Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS. J. Eur. Math. Soc. (JEMS) 14(4), 1275–1330 (2012)

    MathSciNet  MATH  Google Scholar 

  38. Oh, T.: Invariant Gibbs measures and a.s. global well-posedness for coupled KdV systems. Differ. Integral Equ. 22(7–8), 637–668 (2009)

    MathSciNet  MATH  Google Scholar 

  39. Oh, T.: Invariance of the Gibbs measure for the Schrödinger–Benjamin–Ono system. SIAM J. Math. Anal. 41(6), 2207–2225 (2009/10)

  40. Oh, T.: Invariance of the white noise for KdV. Commun. Math. Phys. 292(1), 217–236 (2009)

    MathSciNet  MATH  Google Scholar 

  41. Oh, T.: White noise for KdV and mKdV on the circle. Harmonic and nonlinear partial differential equations, 99–124. RIMS Kokyuroku Bessatu, B18, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010

  42. Oh, T., Quastel, J.: On invariant Gibbs measures conditioned on mass and momentum. J. Math. Soc. Jpn. 65(1), 13–35 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Oh, T., Quastel, J., Valkó, B.: Interpolation of Gibbs measures with white noise for Hamiltonian PDE. J. Math. Pures Appl. (9) 97(4), 391–410 (2012)

    MathSciNet  MATH  Google Scholar 

  44. Oh, T., Richards, G., Thomann, L.: On invariant Gibbs measures for the generalized KdV equations. Dyn. Partial Differ. Equ. 13(2), 133–153 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Olver, P.J.: Unidirectionalization of Hamiltonian waves. Phys. Lett. A. 126(8–9), 501–506 (1988)

    MathSciNet  Google Scholar 

  46. Quastel, J., Valkó, B.: KdV preserves white noise. Commun. Math. Phys. 277(3), 707–714 (2008)

    MathSciNet  MATH  Google Scholar 

  47. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York (1972)

    MATH  Google Scholar 

  48. Rider, B.: On the \(\infty \)-volume limit of the focusing cubic Schrödinger equation. Commun. Pure Appl. Math. 55(10), 1231–1248 (2002)

    MATH  Google Scholar 

  49. Rider, B.: Fluctuations in the thermodynamic limit of focussing cubic Schrödinger. J. Stat. Phys. 113(3–4), 575–594 (2003)

    MATH  Google Scholar 

  50. Simon, B.: Trace ideals and their applications. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, (2005)

  51. Simon, B.: Functional Integration and Quantum Physics, 2nd edn. AMS Chelsea Publishing, Providence, RI 2005. xiv+306pp

  52. Simon, B.: The \(P(\phi )_2\) Euclidean (Quantum) Field Theory. Princeton University Press, Princeton (1974)

    Google Scholar 

  53. Stroock, D.W.: Probability Theory. An Analytic View, 2nd edn. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  54. Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Classics in Mathematics. Springer, Berlin (2006)

    MATH  Google Scholar 

  55. Thomann, L., Tzvetkov, N.: Gibbs measure for the periodic derivative nonlinear Schrödinger equation. Nonlinearity 23(11), 2771–2791 (2010)

    MathSciNet  MATH  Google Scholar 

  56. Tzvetkov, N.: Construction of a Gibbs measure associated to the periodic Benjamin–Ono equation. Probab. Theory Related Fields 146(3–4), 481–514 (2010)

    MathSciNet  MATH  Google Scholar 

  57. Tzvetkov, N., Visciglia, N.: Invariant measures and long time behaviour for the Benjamin–Ono equation. Int. Math. Res. Not. 17, 4679–4714 (2014)

    MATH  Google Scholar 

  58. Tzvetkov, N., Visciglia, N.: Invariant measures and long time behaviour for the Benjamin–Ono equation II. J. Math. Pures Appl. 103, 102–141 (2015)

    MathSciNet  MATH  Google Scholar 

  59. Xu, S.: Invariant Gibbs measure for 3D NLW in infinite volume. Preprint arXiv:1405.3856

  60. Zhou, Y.: Uniqueness of weak solution of the KdV equation. Int. Math. Res. Not. 1997(6), 271–283 (1997)

    MathSciNet  MATH  Google Scholar 

  61. Zimmer, R.: Ergodic theory and semisimple groups. Monographs in mathematics, 81. Birkhäuser, Boston (1984)

Download references

Acknowledgements

R. K. was supported by NSF Grant DMS-1600942. J. M. was supported by a Simons Collaboration grant. M. V. was supported by NSF Grants DMS-1500707 and DMS-1763074. The authors are also grateful to Changxing Miao for hosting us at the Institute of Applied Physics and Computational Mathematics in Beijing, where part of this work was completed. Finally, we would like to thank the anonymous referees for their suggestions and corrections.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, USA

    Rowan Killip & Monica Visan

  2. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, USA

    Jason Murphy

Authors
  1. Rowan Killip
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jason Murphy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Monica Visan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rowan Killip.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Killip, R., Murphy, J. & Visan, M. Invariance of white noise for KdV on the line. Invent. math. 222, 203–282 (2020). https://doi.org/10.1007/s00222-020-00964-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-020-00964-9

Search

Navigation