Weighted semigroup approach for Fokker-Planck-Kolmogorov equations

在这个项目中，我们关注的是非线性抛物型Fokker-Planck-Kolmogorov方程(FPKE)，它是关于测量的微分方程。这样的方程出现在各种科学背景下，例如在统计力学、人口扩散或平均场游戏领域。在这样的区域中，FPKE被用来模拟粒子的空间分布的演变或概率密度的演变。一般类型的FPKES将测量曲线(通常被理解为弱意义上的)的时间导数与定向运动(漂移)加上“随机”、非定向运动(扩散)相关联。在非线性情况下，这些运动不仅依赖于时间和空间，而且依赖于解本身，这使得解的构造成为一项具有挑战性的任务。这种方程在数学上的重要性尤其源于与随机分析的密切联系：FPKE的解等价于自然相关的随机微分方程解，后者用于模拟粒子在确定性和随机力作用下的演化。因此，构造FPKE的解是一项重要的任务。该项目的一个目标是求解一类具有奇异的和无界的漂移和扩散系数的FPKE。在这种情况下，FPKES是函数方程，包括许多来自物理、生物学和地质学的重要偏微分方程组。最近，所谓的半群方法被成功地用于构造某些FPKE的解。这种结果的证明仅限于有界漂移。我们的计划是进一步发展这一方法，以便也包括具有无界漂移的方程。经典方法在关于勒贝格测度的可积函数空间中构造解，而高级方法将基于加权函数空间。我们打算将这种方法也推广到无限维空间。此外，我们将研究这些解的正则化效应，即我们想要证明，在方程扩散部分的适当假设下，即使初始条件是非常奇异的，解也是有界的。作为第二个目标，我们想把我们的发现转移到相关的随机方程上。特别地，在无限维的情况下，这将导致一类随机偏微分方程解的新的存在性结果。在无限维中，与有限维情况相比，对FPKE和SPDEs的解之间的对应关系的理解程度要小得多。通过我们的项目，我们打算在偏微分方程理论和随机分析的界面上在这个方向上取得进展。

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.