 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: $\mathbb{P}(X_t={\tilde{X}}_t\text{ for every }t)=1$. The latter concerns equality of paths on a common space, whereas uniqueness in law concerns equality of probability distributions.