Quasi-Invariance under Flows Generated by Non-Linear PDEs

Abstract

The paper is concerned with the change of probability measures $\mu$ along non-random probability measure valued trajectories $\nu_t$, $t\in [-1,1]$. Typically solutions to non-linear PDEs, modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map $\nu\equiv\nu_0\mapsto\nu_t$ does not exit the state space, for $t\in [-1,0]$ or for $t\in [0,1]$, quasi-invariance of the measure $\mu$ under the map $\nu\mapsto\nu_t$ is established and the Radon-Nikodym derivative of $\mu\circ\nu_t$ with respect to $\mu$ is determined. It is also investigated how Fr\'echet 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 Fr\'echet 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 quasi-invariance is obtained and it is demonstrated how this result can be used in order to derive a corresponding integration by parts formula.