On growth of Sobolev norms for periodic nonlinear Schrödinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics

We prove logarithmic growth bounds on Sobolev norms of the focusing mass-critical NLS and gKdV equations on the torus, which hold almost surely under the focusing Gibbs measure with optimal mass threshold constructed by Oh, Sosoe, and Tolomeo [Invent. Math. 227 (2022), no. 3, 1323–1429]. More precisely, we will establish almost sure growth bounds for solutions $u(t)$ of the equations of the form $$\underset{t\in [-T,T]}{\sup }\parallel u(t){\parallel }_{H^s(\mathbb{T})}{\lesssim }_{s,u_0}\log (2+T)$$ with initial data $u_0\in H^s(\mathbb{T})$ for $s<\frac{1}{2}$. The proof uses a generalisation of Bourgain’s invariant measure argument for measures in a suitable Orlicz space.