= Solution
\b[There is no density; scaling by two gives a law singular to <Wiener measure>.] On $[0,1]$ form the pathwise dyadic sums
$$
Q_n(w)=\sum_{k=1}^{2^n}\bigl(w(k2^{-n})-w((k-1)2^{-n})\bigr)^2.
$$
Under <Wiener measure>, <independent> normal increments give $\mathbb EQ_n=1$ and $\operatorname{var}(Q_n)=2^{1-n}$. Chebyshev's inequality and the <Borel-Cantelli lemma> show $Q_n\to1$ 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.
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