Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 30 / 2 / b / 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, thenThis is indistinguishability of stochastic processes, a statement about coupled sample paths. It is stronger than requiring only equality of their distributions.