On invariant Gibbs measures for the generalized KdV equations

Tadahiro Oh (1) , Geordie Richards (2) , Laurent Thomann (3)

1 Edin. - University of Edinburgh
2 University of Rochester [USA]
3 IECL - Institut Élie Cartan de Lorraine

We consider the defocusing generalized KdV equations on the circle. In particular, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure. In handling a nonlinearity of an arbitrary high degree, we make use of the Hermite polynomials and the white noise functional.

Mots clés: white noise functional; Hermite polynomial; generalized KdV equation; Gibbs measure

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Tadahiro Oh, Geordie Richards, Laurent Thomann. On invariant Gibbs measures for the generalized KdV equations. Dynamics of Partial Differential Equations, 2016, 13 (2), pp.133--153. ⟨10.4310/DPDE.2016.v13.n2.a3⟩. ⟨hal-01203455v2⟩