Quasi-Invariance under Flows Generated by Non-Linear PDEs

Abstract

The paper is concerned with the quasi-invariance of probability measures µ under probability measure valued flows. Typically, solutions to non-linear PDEs modeling spatial development as time progresses generate such flows. In particular, it is shown that under additional conditions on µ, Fréchet differentiability of the solution map of the PDE is sufficient for the quasi-invariance. This result is applied to the Boltzmann equation of cutoff type. The second application is a PDE related to the asymptotic behavior of a Fleming-Viot type system. Here it is demonstrated how the quasi-invariance result can be used to establish a corresponding integration by parts formula.