Stability estimates for singular SDEs and applications

We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $σ$, satisfying Krylov--Röckner type assumptions. We prove several stability estimates, comparing solutions driven by different $(b^i,σ^i)$, 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.

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