Random tensors, propagation of randomness and nonlinear dispersive equations

Abstract: We discuss the random tensor theory, which we develop in recent works collaborated with Andrea R. Nahmod and Haitian Yue, in the context of random data problems for nonlinear dispersive equations. Applying this theory, we establish almost-sure local well-posedness with random data for semilinear Schrodinger equations in the full probabilistic subcritical regime, which is the natural scaling associated with the problem. In 2D this implies the invariance of Gibbs measure with arbitrary power nonlinearity, extending the result of Bourgain (1996).