Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In the first part of the talk, we will discuss the connection of Gibbs measures with the Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation. In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of nonlinear Hamiltonian PDEs. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari. In the second part of the talk, we will explain a general principle that allows us to obtain almost sure global solutions for Hamiltonian PDEs provided that one has a stationary probability measure. In this context, stationarity refers to a solution of the associated Liouville equation. This more general notion replaces the invariance from before. The second part of the talk is joint work with Zied Ammari and Shahnaz Farhat. The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0 Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006)