uniqueness in law

Uniqueness in law (weak uniqueness) is the statement that an SDE has, for each starting point, essentially only one possible probability law for its solution. It does not say the solution path is unique — different weak solutions may live on different probability spaces and trace genuinely different paths — but they all have the same finite-dimensional distributions, hence the same law as random elements of path space. It is the appropriate notion of uniqueness when, as is usually the case in applications, what you measure is a distribution, not a particular trajectory.

Precisely: the SDE dX = b(X) dt + sigma(X) dB has uniqueness in law from a starting point x if any two weak solutions (X, B) on (Omega, F, P) and (X', B') on (Omega', F', P') with X_0 = X'_0 = x satisfy law(X) = law(X') as measures on path space C([0,infinity), R^d). This is genuinely weaker than pathwise uniqueness, which compares X and X' driven by the SAME Brownian motion on the SAME space. Uniqueness in law is exactly the form of uniqueness that the martingale problem delivers: if the martingale problem for the generator L is well-posed, then the law of the diffusion is determined, and the solution is a strong Markov process. Tools that prove it include the martingale-problem well-posedness theorems (Stroock-Varadhan, under continuity and nondegeneracy of a = sigma sigma^T), Girsanov for drift changes, and explicit transition densities.

Why it matters: uniqueness in law is what makes 'the diffusion with generator L' a well-defined object — its semigroup, its generator, and all distributional quantities E[g(X_t)] are unambiguous. The honest interface with the other notions is the Yamada-Watanabe circle: pathwise uniqueness implies uniqueness in law, and weak existence plus pathwise uniqueness gives a unique strong solution; but uniqueness in law alone does NOT imply pathwise uniqueness. A clean caution: nondegeneracy of the diffusion matrix is what usually rescues uniqueness in law for merely-continuous (non-Lipschitz) coefficients, where pathwise uniqueness can fail.