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Quasi-Invariance under Flows Generated by Non-Linear PDEs

Jörg-Uwe Löbus

Abstract

The paper is concerned with the change of probability measures $μ$ along non-random probability measure valued trajectories $ν_t$, $t\in [-1,1]$. Typically solutions to non-linear PDEs, modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map $ν\equivν_0\mapstoν_t$ does not exit the state space, for $t\in [-1,0]$ or for $t\in [0,1]$, quasi-invariance of the measure $μ$ under the map $ν\mapstoν_t$ is established and the Radon-Nikodym derivative of $μ\circν_t$ with respect to $μ$ 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 quasi-invariance is obtained and it is demonstrated how this result can be used in order to derive a corresponding integration by parts formula.

Quasi-Invariance under Flows Generated by Non-Linear PDEs

Abstract

The paper is concerned with the change of probability measures along non-random probability measure valued trajectories , . Typically solutions to non-linear PDEs, modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map does not exit the state space, for or for , quasi-invariance of the measure under the map is established and the Radon-Nikodym derivative of with respect to 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 quasi-invariance is obtained and it is demonstrated how this result can be used in order to derive a corresponding integration by parts formula.

Paper Structure

This paper contains 27 sections, 30 theorems, 386 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation and Basic Hypotheses
  4. Formal definition of flows and trajectories generalizing solutions to PDEs - a motivation
  5. The flow $V_t^+$, $t\in [0,S+1]$
  6. The flow $W^+(t,\cdot)$, $t\in [0,S+1]$, and the trajectory $U^+(t,\cdot)$, $t\in [-1,1]$
  7. State space and direction of inclusion
  8. The measure to follow along the trajectory $U^+$
  9. The generator of the flow $W^+$ and the trajectory $U^+$
  10. Proofs of Lemmas \ref{['Lemma1.0']}-\ref{['Lemma1.1']}
  11. The Change of Measure Formula
  12. General Hypotheses on $\hbox{\boldmath${\mu}$}$ and $A^f$
  13. Specified Hypotheses on $\hbox{\boldmath${\mu}$}$ and $A^f$
  14. General Setup of Subsection \ref{['sec:2:11']}
  15. A class of measures $\hbox{\boldmath${\mu}$}$
  16. ...and 12 more sections

Key Result

Lemma 1.1

The sets $\, {\cal S}^v:=E\cap\{h\cdot\lambda:h\in L^v(D)\}$, $1\le v< \infty$, belong to ${\cal B}(E)$.

Theorems & Definitions (30)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 20 more