Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation

We consider Cauchy problems of some dispersive PDEs with random initial data. In particular, we construct local-in-time solutions to the mean-zero periodic KdV almost surely for the initial data in the support of the mean-zero Gaussian measures on H(s)(T), s > s(0) where s(0) = -11/6 + root 61/6 approximate to -0.5316 <-1/2, by exhibiting nonlinear smoothing under randomization on the second iteration of the Duhamel formulation. We also show that there is no nonlinear smoothing for the dispersionless cubic Szego equation under randomization of initial data.