The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

We consider the focusing energy-critical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in H-x(s)(R-3) x H-x(s-1) (R-3), for any s > 1/2. By randomizing radial initial data in H-x(s)(R-3) x H-x(s-1) (R-3) for s > 5/6, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton that give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the 1st long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.