Abstract: I will discuss recent progress on weak and strong solution theory for SDEs with singular drifts in critical regimes. These results apply to particle systems with strong attracting interactions and, furthermore, yield probabilistic proofs of new analytic results, including an improved upper bound on the constant in the many-particle Hardy inequality. In detail:

1) Weak solutions for form-bounded drifts + divergence-free distributional drifts in the BMO^{-1} space of Koch-Tataru via De Giorgi's method in Lp. This reaches the "ultimate" Lax-Milgram setting and, in a sense, bridges the Eulerian and Lagrangian approaches to describing the physical process of diffusion.

2) Weak solutions for a larger class of weakly form-bounded drifts, including Morrey class drifts, together with related PDE techniques, such as non-local Neumann/Duhamel representations.

3) Strong solutions for form-bounded drifts via the Roeckner-Zhao approach, i.e. their compactness on the Wiener-Sobolev space, but supplemented with some improved PDE estimates.

The talk is based in part on joint papers with R. Madou, Yu.A. Semenov and R. Vafadar.