241016 Fokker-Planck equations (VÜA) (SoSe 2024)

Fokker--Planck equations (FPE) are differential equations for measures. We will reveal the central role FPEs play as a bridge between stochastics and analysis. In itself, they are an interesting class of equations with close connections to PDEs. The most striking aspect, which has sparked substantial recent research interest, is the intimate connection to stochastic analysis: Every stochastic differential equation has a FPE such that the one-dimensional time marginals of solutions of the former solve the latter. After an introduction to the topic, our first cornerstone will be the superposition principle, which reverses this relation. We shall also study the connection to Markov processes.

Teilnahmevoraussetzungen, notwendige Vorkenntnisse

Notwendig: Measure theory, functional analysis, probability theory (Brownian motion, Markov processes); Empfohlen: Basic stochastic analysis (Itô-formula, martingales, SDEs, martingale problem) and PDE theory

Literaturangaben

V. Bogachev, N. Krylov, M. Röckner, S. Shaposhnikov: Fokker-Planck-Kolmogorov equations; D. Stroock, S. Varadhan: Multidimensional Diffusion Processes; D. Trevisan: Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients.