= Solution
<Uniqueness in law> means that any two <weak stochastic solutions> with the same prescribed initial distribution have the same distribution as random elements of the continuous-path space. Their probability spaces and driving <Brownian motions> may differ.
<Pathwise uniqueness> means that if two solutions are defined on the same filtered probability space, driven by the same <Brownian motion>, and have the same initial variable almost surely, then
$$
\boxed{\mathbb P(X_t=\widetilde X_t\text{ for every }t\geq0)=1.}
$$
This is <indistinguishability of stochastic processes>, a statement about coupled sample paths. It is stronger than requiring only equality of their distributions.
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