New examples of probabilistic well-posedness for nonlinear wave equations

We consider fractional wave equations with exponential or arbitrary polynomial nonlinearities. We prove the global well-posedness on the support of the corresponding Gibbs measures. We provide ill-posedness constructions showing that the results are truly super-critical in the considered functional setting. We also present a result in the case of a general randomisation in the spirit of the work by N. Burq and the second author.