Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line

Abstract

We prove that the Gibbs measures $\rho$ for a class of Hamiltonian equations written ${\partial }_{t}u=J(-△u+V^{\prime }(|u{|}^2)u)$ on the real line are invariant under the flow of this equation in the sense that there exist random variables $X(t)$ whose laws are $\rho$ (thus independent from $t$) and such that $t\mapsto X(t)$ is a solution to the above equation. Besides, for all $t$, $X(t)$ is almost surely not in $L^2$ which provides as a direct consequence the existence of weak solutions for initial data not in $L^2$. The proof uses Prokhorov's theorem, Skorohod's theorem, as in the strategy in \cite{burqtzv} and Feynman-Kac's integrals.

28p, we added some remarks over the Schrödinger equation with variable coefficients