Long time behavior of Gross-Pita evskii equation at positive temperature

Anne De Bouard, Arnaud Debussche, Reika Fukuizumi

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


The stochastic Gross-Pitaevskii equation is used as a model to describe Bose-Einstein condensation at positive temperature. The equation is a complex Ginzburg Landau equation with a trapping potential and an additive space-time white noise. Two important questions for this system are the global existence of solutions in the support of the Gibbs measure and the convergence of those solutions to the equilibrium for large time. In this paper, we give a proof of these two results in one space dimension. In order to prove the convergence to equilibrium, we use the associated purely dissipative equation as an auxiliary equation, for which the convergence may be obtained using standard techniques. Global existence is obtained for all initial data, and not almost surely with respect to the invariant measure.

Original languageEnglish
Pages (from-to)5887-5920
Number of pages34
JournalSIAM Journal on Mathematical Analysis
Issue number6
Publication statusPublished - 2018


  • Complex Ginzburg Landau equation
  • Gibbs measure
  • Harmonic potential
  • Stochastic partial differential equations
  • White noise

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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