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Implicit Convolution Fokker-Planck Equations: Extended Feller Convolution

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 325)

Abstract

Fokker-Planck equations are partial differential equations in the transition function of the Markov process. In the evolution equation approach, we re-write partial differential equations as ordinary differential equations in Banach spaces. In particular, an implicit evolution equation is used to re-write the Fokker-Planck equation for a pair of discontinuous Markov processes. In this paper we consider the continuous analogue in the form of two homogeneous Markov processes intertwined by the extended Chapman-Kolmogorov equation. Abstract harmonic analysis techniques are used to extend the Feller convolution. Then the associated Fokker-Planck equations are re-written as an implicit evolution equation expressed in terms of the extended Feller convolution.

Keywords

  • Extended Chapman-Kolmogorov equation
  • Intertwined homogeneous Markov processes

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  • DOI: 10.1007/978-3-030-46079-2_18
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Acknowledgements

W.-S. Lee thanks Professor Jacek Banasiak and Professor Sanne Ter Horst for their decisive, generous and wise support. The support of the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) towards this research is hereby acknowledged by W.-S. Lee. Opinions expressed and conclusions arrived at, are those of the authors and are not necessarily to be attributed to the CoE. Part of the work of W.-S. Lee was also funded by the Claude Leon Foundation and the NWU Postoctoral Fellowship.

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Correspondence to Wha-Suck Lee .

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Lee, WS., Le Roux, C. (2020). Implicit Convolution Fokker-Planck Equations: Extended Feller Convolution. In: Banasiak, J., Bobrowski, A., Lachowicz, M., Tomilov, Y. (eds) Semigroups of Operators – Theory and Applications. SOTA 2018. Springer Proceedings in Mathematics & Statistics, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-030-46079-2_18

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