Weighted semigroup approach for Fokker-Planck-Kolmogorov equations
Final Report Abstract
The main goal of this project, carried out at the Scuola Normale Superiore Pisa in the group of Prof. Franco Flandoli, was to obtain a new existence result for a class of nonlinear Fokker–Planck–Kolmogorov equations (FPKE), which are second-order parabolic partial differential equations (PDEs) for measures. In contrast to previous related results, this class of equations consists of coefficients with unbounded first-order terms, which could not be handled by standard methods before. These equations describe the distributions of random processes which serve as a model for generalized perturbed nonlinear Ornstein–Uhlenbeck processes. It is expected that this new class of processes (as the classical special case, i.e. the well-studied Ornstein–Uhlenbeck process) will become relevant as a model for various phenomena which exhibit a mean-reverting behavior in biology, physics, finance and other branches of science. The method used in this project to prove existence for this class of PDEs is to apply the famous Crandell–Liggett nonlinear semigroup method in weighted L1 -spaces. To do so, one chooses the weight as the density of a symmetrizing measure of the operator appearing in the PDE. This way, it was possible to prove an existence result for bounded initial densities. As a second result, a close connection of the aforementioned PDE solutions to stochastic processes solving a corresponding distribution dependent stochastic differential equation (DDSDE) was established. Such solutions model random evolutions in time, subject to a deterministic drift and a diffusion term (”noise”) which roughly consists of normally distributed time increments. ”Distribution dependent” means the drift and the diffusion depend not only on the current position of the trajectory but also on the distribution among all trajectories at the present time. More precisely, DDSDE-solutions whose distribution at any time t equals the corresponding FPKE-solution at time t were constructed. This allows to transfer properties from the PDE/FPKE-solutions to the DDSDE-solutions and vice versa. For this specific class of PDEs and DDSDES, this connection is new. Finally, it was proven that these DDSDE-solutions satisfy a very natural property, namely a generalized ”nonlinear” Markov property. The essence of processes with this property is that their future evolution depends only on their current state and current distribution, but not on their past states. Due to this new result, the considered class of DDSDE-solutions falls into this class of processes and may therefore be investigated with the very rich tool box of Markov processes in future works.
Publications
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Weighted L1 -semigroup approach for nonlinear Fokker–Planck equations and generalized Ornstein–Uhlenbeck processes.
M. Rehmeier
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Average Dissipation for Stochastic Transport Equations with Lévy Noise. Mathematics of Planet Earth, 45-59. Springer Nature Switzerland.
Flandoli, Franco; Papini, Andrea & Rehmeier, Marco
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Remarks on Regularization by Noise, Convex Integration and Spontaneous Stochasticity. Milan Journal of Mathematics, 92(2), 349-370.
Flandoli, Franco & Rehmeier, Marco
