Quasi-Invariance under Flows Generated by Non-Linear PDEs
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2013
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| _version_ | 1866929528223301632 |
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| author | Löbus, Jörg-Uwe |
| author_facet | Löbus, Jörg-Uwe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1311_0200 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Quasi-Invariance under Flows Generated by Non-Linear PDEs Löbus, Jörg-Uwe Analysis of PDEs 35Qxx, 76P05 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. |
| title | Quasi-Invariance under Flows Generated by Non-Linear PDEs |
| topic | Analysis of PDEs 35Qxx, 76P05 |
| url | https://arxiv.org/abs/1311.0200 |