Fokker-Planck and Kolmogorov (backward) equations can be interpreted as linearisations of the underlying stochastic differential equations (SDE). It turns out that, in particular, on infinite dimensional spaces (i.e. for example if the SDE is a stochastic partial differential equation (SPDE) of evolutionary type), the Fokker-Planck equation is much better to analyze than the Kolmogorov (backward) equation. The reason is that the Fokker-Planck equation is a PDE for measures. Hence e.g. existence of solutions via compactness arguments is easier to show than for PDE on functions. On the other hand uniqueness appears to be much harder to prove.
In this talk we first give a quite elaborate introduction into the relations between S(P)DE, Fokker-Planck and Kolmogorov equations. Subsequently, we shall sketch a new method to prove uniqueness of solutions for Fokker-Planck equations.