The notion of adapted distribution of a stochastic process was introduced in a recent paper of Hoover and Keisler. Here we give a simple characterization of this notion in terms of filtration embeddability. This characterization allows us to show that for a local martingale $M$ for which some ordinary stochastic differential equation $X_t = \int^t_0f(s, X_s) dM_s$ admits sufficient nonuniqueness in law of the solutions $X$, the class of possible joint laws of $(M, X)$ determines the adapted law of $M$.
"A Characterization of Adapted Distribution." Ann. Probab. 15 (4) 1600 - 1611, October, 1987. https://doi.org/10.1214/aop/1176991997