The following formulation of the theorem due Yamada-Watanabe can be found in "On the Existence of Universal Functional Solutions to Classical SDE'S" (Kallenberg, O., 1996).

Assume that weak existence and pathwise uniqueness hold for solutions starting at arbitrary fixed points. Then strong existence and uniqueness in law hold for every initial distribution. Furthermore, there exists a Borel measurable and universally predictable function F(x,w) such that any solution (X,B) satisfies X=F(X(0),B) a.s.

Since weak solutions (for a given initial distribution μ=δx ) might be defined on different probability spaces, then shouldn't the function F depend on the probability space?

I mean, suppose we have a given initial distribution μ=δx, assume two solutions (X,B) and (X′,B)exist on two different filtered probability spaces (Ω,F,Ft,P) and (Ω′,F′,Ft′,P′) such that X(0) and X′(0) are distributed as μ i.e. X(0)=X′(0)=x∈R^d. (This is not incompatible with the pathwise uniqueness since the solutions are defined in different probability spaces).

Hence the solutions should by the previous theorem be X=F(x,B)=X′ a.s. but this two solutions live in different probability spaces!

Am I missing something?

At this point my question reduces to, given a Brownian motion and an initial value, does this suffice to determine the underlying probability space unambiguously?

Thanks in advance.