Nonlinear Fokker-Planck Flows and their Probabilistic Counterparts

Viorel Barbu, Michael Röckner

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English

Springer International Publishing AG
18 June 2024

Differential calculus & equations; Probability & statistics; Stochastics; Mathematical physics

Series: Lecture Notes in Mathematics

This book delves into a rigorous mathematical exploration of the well-posedness and long-time behavior of weak solutions to nonlinear Fokker-Planck equations, along with their implications in the theory of probabilistically weak solutions to McKean-Vlasov stochastic differential equations and the corresponding nonlinear Markov processes. These are widely acknowledged as essential tools for describing the dynamics of complex systems in disordered media, as well as mean-field models. The resulting stochastic processes elucidate the microscopic dynamics underlying the nonlinear Fokker-Planck equations, whereas the solutions of the latter describe the evolving macroscopic probability distributions.

The intended audience for this book primarily comprises specialists in mathematical physics, probability theory and PDEs. It can also be utilized as a one-semester graduate course for mathematicians. Prerequisites for the readers include a solid foundation in functional analysis and probability theory.

By: Viorel Barbu, Michael Röckner
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 2024 ed.
Volume: 2353
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9783031617331
ISBN 10: 3031617339
Series: Lecture Notes in Mathematics
Pages: 214
Publication Date: 18 June 2024
Audience: Professional and scholarly , College/higher education , Undergraduate , Further / Higher Education
Format: Paperback
Publisher's Status: Active

- Introduction.- Existence of nonlinear Fokker–Planck flows.- Time dependent Fokker–Planck equations.- Convergence to equilibrium of nonlinear Fokker–Planck flows.- Markov processes associated with nonlinear Fokker–Planck equations.- Appendix.

Viorel Barbu is professor of Mathematics at Alexandru Ioan Cuza University (Romania) and also member of Romanian Academy and of European Academy of Science. He has published several monographs and textbooks on nonlinear analysis, infinite dimensional optimization, partial differential equations and Navier-Stokes equations with Springer, Academic Press, Kluwer, Birkhauser. Michael Röckner is professor of Mathematics at Bielefeld University (Germany) and a distinguished visiting professor at CAS. He is a member of the Academia Europaea, the Academy of Sciences and Literature, Mainz, and a foreign honorary member of the Romanian Academy. His main areas of research are stochastic analysis, in particular, stochastic partial differential equations, the theory of Dirichlet forms and potential theory. He is a coauthor of several monographs in these fields.

Reviews for Nonlinear Fokker-Planck Flows and their Probabilistic Counterparts

“The authors present in their book a complete analysis of nonlinear Fokker-Planck equations and of McKean-Vlasov stochastic differential equations. The proof of each main result (existence, uniqueness, and long-time behavior) is given with details. The reader will find in this book how classical tools can be applied in the important context of nonlinear equations.” (Alain Brillard, zbMATH 1548.35261, 2024)