= Solution
A <weak solution of a stochastic differential equation> consists of a choice of probability space, <filtration>, driving <Brownian motion> and adapted solution with the specified initial law. The driving noise and the space are part of what may be chosen.
A <strong solution of a stochastic differential equation> is constructed for a prescribed <Brownian motion> and initial value, as a nonanticipating measurable functional of that data; for deterministic initial value it is adapted to the usual augmentation of the Brownian <filtration>. No further independent randomness is required.
<Uniqueness in law> means that any two <weak solutions of a stochastic differential equation> with the same initial distribution have the same law as solution processes. <Pathwise uniqueness> means that two solutions on the same filtered space, driven by the same <Brownian motion> and with the same initial value almost surely, are <indistinguishable>: $\mathbb P(X_t=\widetilde X_t\text{ for every }t)=1$. The latter concerns equality of paths on a common space, whereas <uniqueness in law> concerns equality of probability distributions.
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