Appears in collection : Advances in Nonlinear Analysis and Nonlinear Waves, a conference in honor of Frank Merle

In this talk, we discuss recent developments in the study of the propagation of randomness under the flow of dispersive PDE. In particular, we present a non-technical overview of recent works that led to the resolution of two open problems concerning Gibbs measure invariance for the 2D NLS with arbitrary wave interactions (joint with Yu Deng and Haitian Yue) and for the 3D cubic NLW (joint with Bjoern Bringmann, Yu Deng and Haitian Yue). The first one is proved using the method of random averaging operators, while the second one relies on the theory of random tensors in conjunction with other techniques, such as paracontrolled calculus and heat-wave analysis.