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

TL;DR

This paper proves the invariance of Gibbs measures for certain Hamiltonian equations on the real line and establishes the existence of weak solutions for initial data outside of L^2, using probabilistic and analytical techniques.

Contribution

It demonstrates the invariance of Gibbs measures under Hamiltonian flows on the real line and constructs weak solutions for non-L^2 initial data, extending previous results.

Findings

Gibbs measures are invariant under the flow of the Hamiltonian equations.

Existence of weak solutions for initial data not in L^2.

Application of probabilistic methods like Prokhorov's and Skorohod's theorems.

Abstract

We prove that the Gibbs measures $\rho$ for a class of Hamiltonian equations written $\partial_t u = J (-\triangle u + V'(|u|^2)u)$ 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.