Weighted semigroup approach for Fokker-Planck-Kolmogorov equations

• Applicant: Dr. Marco Rehmeier
• Subject Area: Mathematics
• Term: from 2023 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 517982119

In this project we are concerned with nonlinear parabolic Fokker-Planck-Kolmogorov equations (FPKEs), which are differential equations for measures. Such equations appear in various scientific contexts, for instance in the fields of statistical mechanics, population spreading or mean-field games. In such areas, FPKEs are used to model either the evolution of the spatial distribution of particles or the evolution of a probability density. The generic type of FPKES relates the time derivative of a curve of measures (usually understood in a weak sense) to a directed motion (drift) plus a "random", undirected motion (diffusion). In the nonlinear case, these motions depend not only on time and space, but also on the solution itself, which renders the construction of solutions a challenging task. The mathematical importance of such equations particularly stems from the close connection to stochastic analysis: Solutions to a FPKE are equivalent to solutions to a naturally associated stochastic differential equation, which are used to model the evolution of a particle subject to deterministic and random forces. Hence, constructing solutions to FPKEs is an important task. One aim of the project is to solve a certain class of FPKEs with singular and unbounded drift and diffusion coefficients. In this case, FPKES are equations for functions and include many important partial differential equations from physics, biology and geology. Recently, the so-called semigroup approach was successfully used to construct solutions to certain FPKEs. The proofs for such results are limited to bounded drifts. The plan is to develop this approach further in order to include also equations with unbounded drifts. While the classical approach constructs solutions in a space of integrable functions with respect to Lebesgue measure, the advanced approach will be based on weighted function spaces. We intend to develop this method also for infinite dimensional spaces. Furthermore, we shall investigate regularizing effects for these solutions, i.e. we want to prove that under suitable assumptions on the diffusion part of the equation, solutions are bounded even if the initial condition is a very singular measure. As a second aim, we want to transfer our findings to the associated stochastic equations. In particular, in the infinite-dimensional case this should lead to new existence results for a class of stochastic partial differential equations (SPDEs). In infinite dimensions, the correspondence between solutions to FPKEs and SPDEs is understood to a much lesser extent compared to the finite-dimensional case. By our project, we intend to obtain progress in this direction at the interface of PDE theory and stochastic analysis as well.

• DFG Programme: WBP Fellowship
• International Connection: Italy
• Host: Professor Dr. Franco Flandoli