On the existence of universal functional solutions to classical SDE's
Olav Kallenberg · The Annals of Probability · 1996
Assume that the weak existence and pathwise uniqueness hold for solutions to the equation $dX_t=\sigma(t,X)dB_t + b(t,X)dt$ starting at fixed points. then there exists a Borel measurable function F, such that any solution (X,B) satisfies $X = F(X_0,B)$ a.s. This strengthens a fundamental result of Yamada and Watanbe, where F may depend on the initial distribution $\mu$