Transport properties of Gibbs and Gaussian measures under the flow of Hamiltonian PDEs

Abstract

We study the transport properties of the Gibbs and Gaussian measures on functions/distributions under the flow of Hamiltonian PDEs. In Chapter 2 and 3, we consider the quasi-invariance of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fractional nonlinear Schrodinger equation. For the case of second-order dispersion or greater, we establish an optimal regularity result for the quasi-invariance of these Gaussian measures. Moreover, we obtain an explicit formula for the Radon-Nikodym derivative. In particular, as for the case of fourth-order dispersion, we extend the quasi-invariance results to Sobolev spaces of negative regularity. In Chapter 4, we study focusing Gibbs measures with log-correlated base Gaussian fields on the d-dimensional torus. Using the variational formulation, we prove the non-normalizability of the Gibbs measure with a quartic interaction. We also present the construction of the focusing Gibbs measure with a cubic interaction and non-normalizability of the Gibbs measure for the two-dimensional Zakharov system. In Chapter 5, we consider the Gibbs dynamics for the Zakharov-Yukawa system on the two-dimensional torus T2, namely a Schrodinger-wave system with a Zakharov-type coupling. We construct the Gibbs measure in the weakly nonlinear coupling case and study the invariance of the Gibbs measure under the resulting dynamics.