Energy solutions and (super-)critical singular S(P)DEs

In this talk I will give an overview over recent progress in the understanding of certain singular stochastic dynamics in the so-called scaling-critical and -supercritical regime. Our approach centers around energy solutions as introduced by Jara, Gonçalves and Gubinelli. Leveraging the underlying Markov structure, we prove weak unique solvability of such solutions for certain SPDEs such as critical stochastic surface quasigeostrophic equations and the critical fractional 1-Dstochastic Burgers equation. The solutions are diffusively scaling-invariant and non-Gaussian as processes. We will further point to several more recent robustifications of this method regarding the structure of the underlying equation.

We also develop a theory of energy solutions in the finite-dimensional setting of SDEs with distributional drift and show weak well-posedness of such solutions for drifts which lie in certain supercritical Besov spaces with negative regularity and can have even more singular but local blow-ups.

Moreover, we construct the 1-D self-repelling Brownian polymer (SRBP) which is formally the solution to an SDE with path-dependent distributional drift: the negative gradient of the local time of the solution. This is achieved by rigorously establishing a certain transformation of the SDE to an SPDE that can be uniquely solved in the sense of energy solutions. We then give a dynamic characterization of the SRBP and show that it is superdiffusive and “nowhere self-avoiding”.

Based on joint works with Harry Giles, Nicolas Perkowski and Shyam Popat.