INVARIANT-MEASURES FOR THE 2D-DEFOCUSING NONLINEAR SCHRODINGER-EQUATION

Consider the 2D defocusing cubic NLS iu(t)+Delta u-u\u\(2)=0 with Hami ltonian integral(\del phi\(2)+1/2\phi\(4)). It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing \phi\(4) by : \phi\(4):, is an invariant measure for the appropriatel y modified equation iu(t)+Delta u-[u\u\(2)-2(integral\u\(2)dx)u]=0. Th ere is a well defined flow on the support of the measure. In fact, it is shown that for almost all data phi the solution u, u(0)=phi, satisf ies u(t)-e(it Delta)phi is an element of C(H)s(R), for some s>0. First a result local in time is established and next measure invariance con siderations are used to extend the local result to a global one (cf. [ B2]).