Quantitative quasi-invariance of Gaussian measures below the energy level for the 1D generalized nonlinear Schrödinger equation and application to global well-posedness

We consider the Schrödinger equation on the one dimensional torus with a general odd-power nonlinearity p ≥ 5 , which is known to be globally well-posed in the Sobolev space H^σ(T) , for every σ ≥ 1 , thanks to the conservation and finiteness of the energy. For regularities σ < 1, where this energy is infinite, we explore a globalization argument adapted to random initial data distributed according to the Gaussian measures µ s , with covariance operator (1-Δ)ˢ , for s in a range (sₚ,3/2] . We combine a deterministic local Cauchy theory with the quasi-invariance of Gaussian measures µ s , with additional L q -bounds on the Radon-Nikodym derivatives, to prove that the Gaussian initial data generate almost surely global solutions. These L q -bounds are obtained with respect to Gaussian measures accompanied by a cutoff on a renormalization of the energy; the main tools to prove them are the Boué-Dupuis variational formula and a Poincaré-Dulac normal form reduction. This approach is similar in spirit to Bourgain’s invariant argument [7] and to arecent work by Forlano-Tolomeo in [18].