Stability estimates for singular SDEs and applications

TL;DR

This paper establishes stability estimates for multidimensional SDEs with singular drifts and Sobolev diffusion coefficients, providing new tools for analyzing solution behavior and applications to McKean--Vlasov SDEs and Wong--Zakai theorems.

Contribution

It introduces novel stability estimates for SDEs with singular coefficients, extending previous results to more general settings and applications.

Findings

Stability estimates for solutions with different coefficients

Applications to McKean--Vlasov SDEs and compactness criteria

Wong--Zakai type approximation results

Abstract

We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $\sigma$, satisfying Krylov--R\"ockner type assumptions. We prove several stability estimates, comparing solutions driven by different $(b^i,\sigma^i)$, both for It\^o 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.