Form-boundedness and singular SDEs: recent progress
A talk in the Bielefeld Stochastic Afternoon series by
Damir Kinzebulatov
| Abstract: | The Brownian motion with Hardy-type attracting drift is a well-known counterexample to weak well-posedness of singular SDEs: if the magnitude of the singularity is greater than a certain threshold value, then the corresponding SDE does not have a weak solution. Informally, the attraction to the origin by the drift is too strong. Recently, we proved an almost sharp positive result: if the magnitude of the singularity is strictly less than this threshold value (at least in high dimensions), then the SDE has a weak solution for any form-bounded drift. Form-boundedness is the minimal assumption on the drift such that the corresponding Fokker-Planck-Kolmogorov operator generates a semigroup in some Lp. For instance, weak Ld or Morrey class drifts are form-bounded. As the magnitude of the singularities of form-bounded drifts gets smaller, the theory of singular SDEs becomes more detailed, which includes uniqueness in Krylov-type class, Feller semigroup, heat kernel bounds and strong existence. In my talk, I will survey these and other recent results in the field of singular SDEs. Many of the results that I will discuss are from joint papers with K.R.Madou, Yu.A.Semenov and R.Song. Within the CRC this talk is associated to the project(s): B1 |