Probabilistic perspectives on dispersive partial differential equations

This thesis treats nonlinear dispersive equations with random initial data. First, we study the defocusing energy-critical nonlinear wave equation on Euclidean space. We prove that the scattering mechanism, which is well-understood for smooth initial data, is stable under rough and random perturbations. The main ingredients are Bourgain's bush argument, flux estimates, and a wave packet decomposition of the random linear evolution. Second, we study the three-dimensional wave equation with a Hartree nonlinearity. The main theorem proves the existence and invariance of the Gibbs measure. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. The argument combines techniques from several areas of mathematics, such as dispersive equations, harmonic analysis, and random matrix theory