Electronic Journal of Probability

Stochastic complex Ginzburg-Landau equation with space-time white noise

Masato Hoshino, Yuzuru Inahama, and Nobuaki Naganuma

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We study the stochastic cubic complex Ginzburg-Landau equation with complex-valued space-time white noise on the three dimensional torus. This nonlinear equation is so singular that it can only be understood in a renormalized sense. In the first half of this paper we prove local well-posedness of this equation in the framework of regularity structure theory. In the latter half we prove local well-posedness in the framework of paracontrolled distribution theory.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 104, 68 pp.

Received: 20 February 2017
Accepted: 11 November 2017
First available in Project Euclid: 15 December 2017

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 82C28: Dynamic renormalization group methods [See also 81T17]

stochastic partial differential equation complex Ginzburg-Landau equation regularity structure paracontrolled distribution renormalization

Creative Commons Attribution 4.0 International License.


Hoshino, Masato; Inahama, Yuzuru; Naganuma, Nobuaki. Stochastic complex Ginzburg-Landau equation with space-time white noise. Electron. J. Probab. 22 (2017), paper no. 104, 68 pp. doi:10.1214/17-EJP125. https://projecteuclid.org/euclid.ejp/1513349792

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  • [AK02] Igor S. Aranson and Lorenz Kramer. The world of the complex ginzburg-landau equation. Rev. Mod. Phys., 74(1):99–143, 2002.
  • [BCD11] Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin. Fourier analysis and nonlinear partial differential equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2011.
  • [BBF15] Ismaël Bailleul, Frédéric Bernicot, and Dorothee Frey. Higher order paracontrolled calculus, 3d-PAM and multiplicative Burgers equations, 2015. arXiv:1506.08773
  • [BS04a] Marc Barton-Smith. Global solution for a stochastic Ginzburg-Landau equation with multiplicative noise. Stochastic Anal. Appl., 22(1):1–18, 2004.
  • [BS04b] Marc Barton-Smith. Invariant measure for the stochastic Ginzburg Landau equation. NoDEA Nonlinear Differential Equations Appl., 11(1):29–52, 2004.
  • [BK16] Nils Berglund and Christian Kuehn. Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions. Electron. J. Probab., 21:Paper No. 18, 1–48, 2016.
  • [CC13] Rémi Catellier and Khalil Chouk. Paracontrolled distributions and the 3-dimensional stochastic quantization equation. Ann. Probab., in press.
  • [FH14] Peter K. Friz and Martin Hairer. A course on rough paths. Universitext. Springer, Cham, 2014.
  • [FH17] Tadahisa Funaki and Masato Hoshino. A coupled KPZ equation, its two types of approximations and existence of global solutions. J. Funct. Anal., 273:1165–1204, 2017.
  • [GIP15] Massimiliano Gubinelli, Peter Imkeller, and Nicolas Perkowski. Paracontrolled distributions and singular PDEs. Forum Math. Pi, 3:e6, 75, 2015.
  • [GP17] Massimiliano Gubinelli and Nicolas Perkowski. KPZ Reloaded. Comm. Math. Phys., 349(1):165–269, 2017.
  • [Hai02] Martin Hairer. Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields, 124(3):345–380, 2002.
  • [Hai14] Martin Hairer. A theory of regularity structures. Invent. Math., 198(2):269–504, 2014.
  • [Hos16] Masato Hoshino. KPZ equation with fractional derivatives of white noise. Stoch. Partial Differ. Equ. Anal. Comput., 4(4):827–890, 2016.
  • [Hos17a] Masato Hoshino. Paracontrolled calculus and Funaki-Quastel approximation for the KPZ equation. Stochastic Process. Appl., in press.
  • [Hos17b] Masato Hoshino. Global well-posedness of complex Ginzburg-Landau equation with a space-time white noise. Ann. Inst. Henri Poincaré Probab. Stat., in press.
  • [Itô52] Kiyosi Itô. Complex multiple Wiener integral. Jap. J. Math., 22:63–86, 1952.
  • [KS04] Sergei Kuksin and Armen Shirikyan. Randomly forced CGL equation: stationary measures and the inviscid limit. J. Phys. A, 37(12):3805–3822, 2004.
  • [Kun90] Hiroshi Kunita. Stochastic flows and stochastic differential equations, volume 24 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990.
  • [Kup16] Antti Kupiainen. Renormalization group and stochastic PDEs. Ann. Henri Poincaré, 17(3):497–535, 2016.
  • [MW17] Jean-Christophe Mourrat and Hendrik Weber. The dynamic $\Phi ^4_3$ model comes down from infinity. Comm. Math. Phys., 356(3):673–753, 2017..
  • [Nua06] David Nualart. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, Berlin, second edition, 2006.
  • [Oda06] Cyril Odasso. Ergodicity for the stochastic complex Ginzburg-Landau equations. Ann. Inst. H. Poincaré Probab. Statist., 42(4):417–454, 2006.
  • [PG11] Xueke Pu and Boling Guo. Momentum estimates and ergodicity for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. J. Differential Equations, 251(7):1747–1777, 2011.
  • [Yan04] Desheng Yang. The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise. J. Math. Phys., 45(11):4064–4076, 2004.
  • [ZZ15] Rongchan Zhu and Xiangchan Zhu. Three-dimensional Navier-Stokes equations driven by space-time white noise. J. Differential Equations, 259(9):4443–4508, 2015.