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Quasi-invariance under flows generated by non-linear PDEs
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2024 (English)In: Analysis and Applications, ISSN 0219-5305, E-ISSN 1793-6861 , Vol. 22, no 01, p. 179-277Article in journal (Refereed) Epub ahead of print
Abstract [en]

The paper is concerned with the change of probability measures μμ along non-random probability measure-valued trajectories νtνt, t∈[−1,1]t∈[−1,1]. 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 ν≡ν0↦νtν≡ν0↦νt does not exit the state space, for t∈[−1,0]t∈[−1,0] or for t∈[0,1]t∈[0,1], the Radon–Nikodym derivative dμ∘νt/dμdμ∘νt/dμ 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.

Place, publisher, year, edition, pages
WORLD SCIENTIFIC PUBL CO PTE LTD , 2024. Vol. 22, no 01, p. 179-277
Keywords [en]
Non-linear PDE; change of measure; quasi-invariance; Boltzmann equation; Fleming-Viot system
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-198970DOI: 10.1142/S0219530523500264ISI: 001082522300001OAI: oai:DiVA.org:liu-198970DiVA, id: diva2:1810034
Available from: 2023-11-06 Created: 2023-11-06 Last updated: 2024-02-22

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