Stochastic differential equations Eulerian correspondence

This is a direct result of forcing the stochastic differential equations to correspond to the Eulerian PDF. From (6.152), it follows that... 

Thus, correspondence between the notional-particle system and the Eulerian PDF of the flow requires agreement at the moment level. In particular, it requires that (U(x, /)) = (U (r) X (0 = x) and ((x, t)) = (0 (r) X (O = x). It remains then to formulate stochastic differential equations for the notional-particle system which yield the desired correspondence. 

Note that Lagrangian correspondence does not automatically imply Eulerian correspondence, since the latter also requires that /x (x t) must be a uniform distribution for all t. Nevertheless, Lagrangian correspondence is the preferable choice, and is obtained by working directly with the stochastic differential equations. 

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