Sharp solvability for singular SDEs

TL;DR

This paper establishes positive solvability results for a class of singular stochastic differential equations with inverse-square type drifts, using a novel $L^p$ De Giorgi method, extending known critical thresholds.

Contribution

It proves solvability for singular SDEs with form-bounded drifts at the critical inverse-square threshold, a significant extension of existing theory.

Findings

Solvability holds for inverse-square and critical Ladyzhenskaya-Prodi-Serrin class drifts.

The $L^p$ De Giorgi method is effective for analyzing singular SDEs.

The critical threshold for solvability is precisely characterized.

Abstract

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 $L^p$ variant of De Giorgi's method.