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Invariance of the White Noise for KdV

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Abstract

We prove the invariance of the mean 0 white noise for the periodic KdV. First, we show that the Besov-type space \({\widehat{b}^s_{p,\infty}}\) , sp < −1, contains the support of the white noise. Then, we prove local well-posedness in \({\widehat{b}^s_{p, \infty}}\) for p = 2 + , \({s = -\frac{1}{2}+}\) such that sp < −1. In establishing the local well-posedness, we use a variant of the Bourgain spaces with a weight. This provides an analytical proof of the invariance of the white noise under the flow of KdV obtained in Quastel-Valko [21].

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References

  1. Bejenaru I., Tao T.: Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation. J. Funct. Anal. 233, 228–259 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. GAFA. 3, 209–262 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bourgain J.: Periodic Korteweg-de Vries equation with measures as initial data. Sel. Math., New Ser. 3, 115–159 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Bourgain J.: On the Cauchy and invariant measure problem for the periodic Zakharov system. Duke Math. J. 76, 175–202 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Burq, N., Tzvetkov, N.: Invariant measure for a three dimensional nonlinear wave equation. Int. Math. Res. Not. (2007), no. 22, Art. ID rnm108, 26pp

  7. Cambronero S., McKean H.P.: The ground state eigenvalue of Hill’s equation with white noise potential. Comm. Pure Appl. Math. 52(10), 1277–1294 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Christ M., Colliander J., Tao T.: Asymptotics, frequency modulation, and low-regularity illposedness of canonical defocusing equations. Amer. J. Math. 125(6), 1235–1293 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.: Sharp Global Well-Posedness for KdV and Modified KdV on \({\mathbb{R} }\) and \({\mathbb{T}}\) . J. Amer. Math. Soc. 16(3), 705–749 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fernique M.X.: Intégrabilité des Vecteurs Gaussiens. Comptes Rendus, Séries A 270, 1698–1699 (1970)

    MATH  MathSciNet  Google Scholar 

  11. Ginibre J., Tsutsumi Y., Velo G.: On the Cauchy Problem for the Zakharov System. J. Funct. Anal. 151, 384–436 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gross, L.: Abstract Wiener spaces. Proc. 5th Berkeley Sym. Math. Stat. Prob. 2 Berkeley CA: Univ Calif. Press, 1965, 31–42

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Kenig C., Ponce G., Vega L.: A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9(2), 573–603 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kenig C., Ponce G., Vega L.: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106(3), 617–633 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kenig C., Ponce G., Vega L.: Quadratic forms for the 1-D semilinear Schrödinger equation. Trans. Amer. Math. Soc. 348, 3323–3353 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kishimoto N.: Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity \({\overline{u}^2}\) . Comm. Pure Appl. Anal. 7(5), 1123–1143 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kuo, H.: Gaussian Measures in Banach Spaces. Lec. Notes in Math. 463, New York: Springer-Verlag, 1975

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

    Google Scholar 

  20. Oh, T.: Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system. Preprint, available at http://arXiv.org/abs/0904.2816v1[math.AP], 2009

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

    Article  MATH  ADS  Google Scholar 

  22. Roynette B.: Mouvement brownien et espaces de Besov. Stochastics Stoch. Rep. 43, 221–260 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  23. Tzvetkov N.: Invariant measures for the nonlinear Schrödinger equation on the disc. Dyn. Partial Differ. Equ. 3(2), 111–160 (2006)

    MathSciNet  Google Scholar 

  24. Tzvetkov N.: Invariant measures for the defocusing Nonlinear Schrödinger equation (Mesures invariantes pour l’équation de Schrödinger non linéaire). Ann. l’Inst. Fourier 58, 2543–2604 (2008)

    MATH  MathSciNet  Google Scholar 

  25. Zhidkov, P.: Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory. Lec. Notes in Math. 1756, Berlin-Heidleperg-NewYork:Springer-Verlag, 2001

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  1. Department of Mathematics, University of Toronto, 40 St. George St, Rm 6290, Toronto, ON, M5S 2E4, Canada

    Tadahiro Oh

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  1. Tadahiro Oh
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Correspondence to Tadahiro Oh.

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Communicated by P. Constantin

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Oh, T. Invariance of the White Noise for KdV. Commun. Math. Phys. 292, 217–236 (2009). https://doi.org/10.1007/s00220-009-0856-7

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