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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.
List of contents
- 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.
Report
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)
