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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Lee, W.-S., Sauer, N.: Intertwined Markov processes: the extended Chapman-Kolmogorov equation. Proc. R. Soc. Edinb. A 148, 123–131 (2018)
Bobrowski, A.: Functional Analysis for Probability and Stochastic Processes. Cambridge University Press, Cambridge (2005)
Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II, 2nd edn. Wiley Ser. Pro. Wiley, New York (1971)
Lee, W.-S., Sauer, N.: Intertwined evolution operators. Semigroup Forum 94, 204–228 (2017)
Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. AMS Colloq. Publ. 31, AMS (2000)
Sauer, N., Banasiak, J., Lee, W.-S.: Causal relations in support of implicit evolution equations. Bull. South Ural State Univ. Ser.: Math. Model. Program. Comput. Softw. 11, 85–102 (2018)
Böttcher, B., Schilling, R., Wang, J.: Lévy Matters III. Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Lecturer Notes in Mathematics, vol. 2099. Springer, Berlin (2013)
Sauer, N.: Empathy theory and the Laplace transform. In: Linear Operators. Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warszawa, vol. 38, pp. 325–338 (1997)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-46079-2_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-46078-5
Online ISBN: 978-3-030-46079-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)