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There is no density; scaling by two gives a law singular to Wiener measure. On $[0,1]$ form the pathwise dyadic sumsUnder Wiener measure, independent normal increments give $\mathbb{E}{Q}_n=1$ and $\mathrm{var}(Q_n)=2^{1-n}$. Chebyshev's inequality and the Borel-Cantelli lemma show $Q_n\to 1$ almost surely, because the variances are summable. For $2B$, every sum is four times its Brownian counterpart and hence tends to four almost surely. The event that the limit equals one separates the two laws. Unlike the linear-drift case, this obstruction already occurs on a finite interval: it is the Pathwise quadratic variation distinguishes Brownian speeds argument.

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.