Almost Sure Global Well-posedness for the Fractional Cubic Schrodinger Equation on the Torus

In a previous paper, we proved that the 1-d periodic fractional Schrodinger equation with cubic nonlinearity is locally well-posed in H-s for s > 1 - alpha/2 and globally well-posed for s > 10 alpha - 1/12. In this paper we define an invariant probability measure mu on H-s for s < alpha - 1/2, so that for any epsilon > 0 there is a set Omega subset of H-s such that mu(Omega(c)) < epsilon and the equation is globally well-posed for initial data in Omega. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense for 1-alpha/2 < alpha - 1/2, i.e., alpha > 2/3 in an almost sure sense.