Quasi-Invariance under Flows Generated by Non-Linear PDEs

Abstract

The paper is concerned with the change of probability measures [Formula: see text] along non-random probability measure-valued trajectories [Formula: see text], [Formula: see text]. Typically solutions to non-linear partial differential equations (PDEs), modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map [Formula: see text] does not exit the state space, for [Formula: see text] or for [Formula: see text], the Radon–Nikodym derivative [Formula: see text] is determined. It is also investigated how Fréchet differentiability of the solution map of the PDE can contribute to the existence of this Radon–Nikodym derivative. The first application is a certain Boltzmann type equation. Here, the Fréchet derivative of the solution map is calculated explicitly and quasi-invariance is established. The second application is a PDE related to the asymptotic behavior of a Fleming–Viot type particle system. Here, it is demonstrated how quasi-invariance can be used in order to derive a corresponding integration by parts formula.