= 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.
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