The attracting inverse-square drift provides a prototypical counterexample to solvability of singular SDEs: if the coefficient of the drift is larger than a certain critical value, then no weak solution exists. We prove a positive result on solvability of singular SDEs where this critical value is attained from below (up to strict inequality) for the entire class of form-bounded drifts. This class contains e.g. the inverse-square drift, the critical Ladyzhenskaya-Prodi-Serrin class. The proof is based on a variant of De Giorgi’s method.
The research of D.K. is supported by the NSERC (grant RGPIN-2017-05567).
"Sharp solvability for singular SDEs." Electron. J. Probab. 28 1 - 15, 2023. https://doi.org/10.1214/23-EJP957