SPASS - Probability, Stochastic Analysis and Statistics Seminar

Transport of Gaussian measures under the flow of Hamiltonian PDEs: quasi-invariance and singularity

by Leonardo Tolomeo (University of Edinburgh)

In this talk, we consider the Cauchy problem for the fractional NLS with cubic nonlinearity (FNLS), posed on the one-dimensional torus T, subject to initial data distributed according to a family of Gaussian measures.  

We first discuss how the flow of Hamiltonian equations transports these Gaussian measures. When the transported measure is absolutely continuous with respect to the initial measure, we say that the initial measure is quasi-invariant. 

In the high-dispersion regime, we exploit quasi-invariance to build a (unique) global flow for initial data with negative regularity, in a regime that cannot be replicated by the deterministic (pathwise) theory.  

In the 0-dispersion regime, we discuss the limits of this approach, and exhibit a sharp transition from quasi-invariance to singularity, depending on the regularity of the initial measure. 

This is based on joint works with J. Forlano (UCLA/University of Edinburgh) and with J. Coe (University of Edinburgh).