Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T13:09:31.220Z Has data issue: false hasContentIssue false

On the Cameron–Martin theorem and almost-sure global existence

Published online by Cambridge University Press:  17 December 2015

Tadahiro Oh
Affiliation:
School of Mathematics, University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK ([email protected])
Jeremy Quastel
Affiliation:
Departments of Mathematics and Statistics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada ([email protected]) and School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA

Abstract

In this paper we discuss various aspects of invariant measures for nonlinear Hamiltonian partial differential equations (PDEs). In particular, we show almost-sure global existence for some Hamiltonian PDEs with initial data of the form ‘a smooth deterministic function + a rough random perturbation’ as a corollary to the Cameron–Martin theorem and known almost-sure global existence results with respect to Gaussian measures on spaces of functions.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bényi, Á. and Oh, T., Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math. 228(5) (2011), 29432981.Google Scholar
2. Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II, Geom. Funct. Analysis 3(3) (1993), 209262.Google Scholar
3. Bourgain, J., Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys. 166 (1994), 126.Google Scholar
4. Bourgain, J., On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J. 76 (1994), 175202.Google Scholar
5. Bourgain, J., Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys. 176(2) (1996), 421445.CrossRefGoogle Scholar
6. Bourgain, J., Invariant measures for the Gross-Pitaevskii equation, J. Math. Pures Appl. 76(8) (1997), 649702.CrossRefGoogle Scholar
7. Bourgain, J., Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not. 1998(5) (1998), 253283.Google Scholar
8. Bourgain, J., Nonlinear Schrödinger equations, in Hyperbolic equations and frequency interactions, IAS/Park City Mathematics Series, Volume 5, pp. 3157 (American Mathematical Society/IAS/Park City Mathematics Institute, 1999).CrossRefGoogle Scholar
9. Bourgain, J., Invariant measures for NLS in infinite volume, Commun. Math. Phys. 210(3) (2000), 605620.Google Scholar
10. Bourgain, J., A remark on normal forms and the ‘I-method’ for periodic NLS, J. Analysis Math. 94 (2004), 125157.Google Scholar
11. Burq, N. and Tzvetkov, N., Invariant measure for a three dimensional nonlinear wave equation, Int. Math. Res. Not. 2007(22) (2007), Art. ID rnm108.Google Scholar
12. Burq, N. and Tzvetkov, N., Random data Cauchy theory for supercritical wave equations, I: local theory, Invent. Math. 173(3) (2008), 449475.CrossRefGoogle Scholar
13. Burq, N. and Tzvetkov, N., Random data Cauchy theory for supercritical wave equations, II: a global existence result, Invent. Math. 173(3) (2008), 477496.CrossRefGoogle Scholar
14. Burq, N. and Tzvetkov, N., Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. 16(1) (2014), 130.Google Scholar
15. Burq, N., Thomann, L. and Tzvetkov, N., Global infinite energy solutions for the cubic wave equation, Bull. Soc. Math. France 143(2) (2015), 301313.Google Scholar
16. Cameron, R. and Martin, W., Transformations of Wiener integrals under translations, Annals Math. (2) 45 (1944), 386396.CrossRefGoogle Scholar
17. Colliander, J. and Oh, T., Almost sure well-posedness of the periodic cubic nonlinear Schrödinger equation below L 2(T), Duke Math. J. 161(3) (2012), 367414.Google Scholar
18. Prato, G. Da, An introduction to infinite-dimensional analysis, Universitext (Springer, 2006).Google Scholar
19. Deng, Y., Invariance of the Gibbs measure for the Benjamin–Ono equation, J. Eur. Math. Soc. 17(5) (2015), 11071198.Google Scholar
20. Suzzoni, A.-S. de, Invariant measure for the cubic wave equation on the unit ball of R3 , Dynam. PDEs 8(2) (2011), 127147.Google Scholar
21. Suzzoni, A.-S. de, Wave turbulence for the BBM equation: stability of a Gaussian statistics under the flow of BBM, Commun. Math. Phys. 326(3) (2014), 773813.CrossRefGoogle Scholar
22. Feldman, J., Equivalence and perpendicularity of Gaussian processes, Pac. J. Math. 8 (1958), 699708.Google Scholar
23. Freidlin, M. and Wentzell, A., Random perturbations of dynamical systems, 3rd edn, Grundlehren der Mathematischen Wissenschaften, Volume 260 (Springer, 2012).Google Scholar
24. Gross, L., Abstract Wiener spaces, in Proc. 5th Berkeley Symposium on Mathematical Statistics and Probability, Volume 2, pp. 3142 (University of California Press, Berkeley, CA, 1965).Google Scholar
25. Hájek, J., On a property of normal distribution of any stochastic process, Czech. Math. J. 8(83) (1958), 610618 (in Russian).Google Scholar
26. Kakutani, S., On equivalence of infinite product measures, Annals Math. (2) 49 (1948). 214224.Google Scholar
27. Kappeler, T. and Topalov, P., Global wellposedness of KdV in H- 1(,), Duke Math. J. 135(2) (2006), 327360.Google Scholar
28. Kuo, H., Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Volume 463 (Springer, 1975).Google Scholar
29. J. Lebowitz, H. Rose and Speer, E., Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys. 50(3) (1988), 657687.Google Scholar
30. McKean, H. P., Statistical mechanics of nonlinear wave equations, IV: cubic Schrödinger, Commun. Math. Phys. 168(3) (1995), 479491 (Erratum, Commun. Math. Phys. 173(3) (1995), 675).CrossRefGoogle Scholar
31. Nahmod, A., Oh, T. ,Rey-Bellet, L.and Staffilani, G., Invariant weighted Wiener measures and almost-sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc. 14 (2012), 12751330.Google Scholar
32. Nahmod, A., Pavlović, N. and Staffilani, G., Almost sure existence of global weak solutions for super-critical Navier–Stokes equations, SIAM J. Math. Analysis 45(6) (2013), 34313452.Google Scholar
33. Oh, T., Gibbs measures, Invariant and global, a.s. well-posedness for coupled KdV systems, Diff. Integ. Eqns 22(7–8) (2009), 637668.Google Scholar
34. Oh, T., Invariance of the white noise for KdV, Commun. Math. Phys. 292(1) (2009), 217236.Google Scholar
35. Oh, T., Periodic stochastic Korteweg-de Vries equation with the additive space-time white noise, Analysis PDEs 2(3) (2009), 281304.Google Scholar
36. Oh, T., Invariance of the Gibbs measure for the Schrödinger–Benjamin–Ono system, SIAM J. Math. Analysis 41(6) (2009), 22072225.Google Scholar
37. Oh, T., White noise for KdV and mKdV on the circle, RIMS Kôkyûroku Bessatsu B18 (2010), 99124.Google Scholar
38. Oh, T., Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation, Funkcial. Ekvac. 54(3) (2011), 335365.Google Scholar
39. Oh, T. and Sulem, C., On the one-dimensional cubic nonlinear Schrödinger equation below L2, Kyoto J. Math. 52(1) (2012), 99115.Google Scholar
40. Oh, T., Quastel, J.and Valkó, B., Interpolation of Gibbs measures with white noise for Hamiltonian PDE, J. Math. Pures Appl. 97(4) (2012), 391410.Google Scholar
41. J. Quastel, B. Valkó, KdV preserves white noise, Commun. Math. Phys. 277(3) (2008), 707714.CrossRefGoogle Scholar
42. Richards, G., Invariance of the Gibbs measure for the periodic quartic gKdV, Annales Inst. H. Poincaré Analyse Non Linéaire, in the press.Google Scholar
43. Shigekawa, I., Stochastic analysis, Translations of Mathematical Monographs, Volume 224 (American Mathematical Society, Providence, RI, 2004).Google Scholar
44. Tzvetkov, N., Invariant measures for the nonlinear Schrödinger equation on the disc, Dynam. PDEs 3(2) (2006), 111160.Google Scholar
45. Tzvetkov, N., Invariant measures for the defocusing nonlinear Schrödinger equation, Annales Inst. Fourier 58 (2008), 25432604.Google Scholar
46. Tzvetkov, N., Construction of a Gibbs measure associated to the periodic Benjamin–Ono equation, Prob. Theory Relat. Fields 146 (2010), 481514.Google Scholar
47. Varadhan, S. R. S., Asymptotic probabilities and differential equations, Commun. Pure Appl. Math. 19 (1966), 261286.CrossRefGoogle Scholar
48. Varadhan, S. R. S., Large deviations, Lecture notes, New York University (available at www.math.nyu.edu/faculty/varadhan/LDP.html, 2010).Google Scholar