Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 5 / c / i / Solution

The assertion is false. For a common sequence, each term of which is a refining deterministic partition of an interval, standard Brownian paths have quadratic variation $t$ almost surely, whereas the paths $t\mapsto B_{2t}$ have quadratic variation $2t$ almost surely. These two path properties define disjoint measurable subsets of $C([0,1])$, so the two laws are mutually singular measures. In particular, the law of $B_{2\cdot }$ is not absolutely continuous with respect to Wiener measure. This is the Pathwise quadratic variation distinguishes Brownian speeds argument.