Stability estimates for singular SDEs and applications

We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $\sigma$, satisfying Krylov–Röckner type assumptions. We prove several stability estimates, comparing solutions driven by different $(b^i,{\sigma }^i)$, both for Itô and Stratonovich SDEs, possibly depending on negative Sobolev norms of the difference $b^1-b^2$. We then discuss several applications of these results to McKean–Vlasov SDEs, criteria for strong compactness of solutions and Wong–Zakai type theorems.