Tadahiro Oh (Edinburgh)
On the transport property of Gaussian measures under Hamiltonian PDE dynamics
Abstract:
Transport properties of Gaussian measures under different
transformations have been studied in probability theory. In this
talk, we discuss transport properties of Gaussian measures on periodic
functions under nonlinear Hamiltonian PDEs such as the nonlinear
Schrodinger equations and nonlinear wave equations.
Lebowitz-Rose-Speer '88, Bourgain '94, and McKean '95 initiated the
study of invariant Gibbs measures for dispersive Hamiltonian PDEs. In
the first part of the talk, we give a review on the construction of
invariant Gibbs measures and discuss how it lead to a recent
development of probabilistic construction of solutions in late 2000’s.
In the second part, we discuss the quasi-invariance property of
Gaussian measures on Sobolev spaces under certain dispersive
Hamiltonian PDEs. We also discuss the importance of dispersion in this
quasi-invariance result by showing that the transported measure and
the original Gaussian measure are mutually singular when we turn off
dispersion. The second part of the talk is based on a joint work with
Nikolay Tzvetkov (Universite Cergy-Pontoise) and Philippe Sosoe
(Harvard University).