Sufficient conditions for the existence and uniqueness of a weak solution of a stochastic differential equation $$dX_t = \mu(X_t)dt + \sigma dZ_t$$ are known. Loosely speaking, Lipschitz conditions along with finite variation of $\mu$ and $\sigma$ are sufficient. I'm interested in a slightly different problem: given a solution, are the parameters unique?
Assume that $x$ follows some stochastic process $$dX_t = \mu(X_t)dt + \sigma dZ_t$$ where $\mu(\cdot)$ is unknown but $\sigma$ is known; and $Z_t$ is a Brownian motion. Given an initial distribution $f_0(x)$ and a final distribution $f_T(x)$ (observed at time $T$), in what sense is $\mu$ unique? Can there be multiple $\mu$, differing on a set of positive measure that generate the same final distribution?
I tried to proceed by contradiction and assume there exist two solutions, $\mu_1$ and $\mu_2$, which generate solutions $f_1$ and $f_2$ that agree at $t = 0, T$. It is sufficient to consider the case where $f_1$ and $f_2$ are equal no where on $(0, T)$ and only at $t=0, T$, as otherwise you can consider the problem starting at the point where they are equal. Furthermore, they cannot be equal everywhere, as this violates the existence of a unique weak solution given parameters (the first point I mentioned above). However, no obvious contradiction emerged.
I didn't find a solution looking through Stroock & Varadhan's textbook and Rogers & Williams' textbook, but it seems like the statement ought to be true (no obvious counterexample has surfaced).
The answer is no.
Take two bridges of your liking such as a Brownian Bridge (with a sigma on hte brownian part of the diuffusion equation) which starts 0 and ends at 0 at time t=1, then take any other bridge with the same start and end point and with the same Brownian parameter in the diffusion (to construct alternative bridge you can use the h-transform (you seem to be a reader of Rogers and Williams textbook you can have a look there where this subject it is treated unless mistaken) where you can see that such a transform do not alter the diffusion part only the drift part and you are done.
Then you have in the simplest case regarding $f_0$ and $f_1$ (i.e. dirac masses) a systematic way to have counterexamples.
Regards