A Brownian SDE with merely measurable, Markovian drift is claimed to have a unique strong solution and H-C regular solution map whenever its Girsanov density has a finite L^{1+ε} moment.
Abstract Wiener spaces
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Strong solutions of SDE's with rough coefficients
A Brownian SDE with merely measurable, Markovian drift is claimed to have a unique strong solution and H-C regular solution map whenever its Girsanov density has a finite L^{1+ε} moment.