Lukas Grafner (University of Warwick) – Energy solutions and singular SDEs with distributional drift

In this talk I will give an overview over recent applications of Energy solutions from the field of singular SPDEs to singular SDEs with distributional drift.

In the first part, I will present weak well-posedness results for Energy solutions in the case of, up to perturbations, divergence-free drifts which lie in certain scaling-supercritical function spaces and can have even more singular but local blow-ups.

In the second part and depending on time, I will present a recent construction of the 1-D self-repelling Brownian polymer (SRBP), which is formally the solution to an SDE with a certain path-dependent distributional drift. Our approach exploits the fact that the SDE is formally in one-to-one correspondence with a singular SPDE that can be solved uniquely in the sense of Energy solutions. We then give a unique dynamic characterization of the law of the SRBP and show that the process is superdiffusive and not self-avoiding.

Based on joint works with Nicolas Perkowski and Harry Giles.