On the uniqueness of probability measure solutions to Liouville's equation of Hamiltonian PDEs

TL;DR

This paper establishes the uniqueness of probability measure solutions to Liouville's equation for Hamiltonian PDEs in infinite-dimensional spaces, extending finite-dimensional results and broadening applicable velocity fields.

Contribution

It provides a new uniqueness theorem for measure solutions in infinite-dimensional Hilbert spaces, generalizing previous finite-dimensional and specific velocity field results.

Findings

Proves uniqueness of measure solutions for Hamiltonian PDEs in infinite dimensions

Extends finite-dimensional Liouville equation results to infinite-dimensional settings

Utilizes projective and probabilistic methods for analysis

Abstract

In this paper, we give a uniqueness result to a transport equation fulfilled by probability measure on a infinite dimensional Hilbert space. Main arguments are based on projective aspects and a probabilistic representation of the solutions. It extends the work of Maniglia, which concerns the finite dimensional case and the work of Ammari and Nier, for a wider class of velocity field.