= Solution
\b[There is no density on the full infinite path space; the two laws are mutually singular.] Under <Wiener measure>, $B_t/t\to0$ almost surely. Under the shifted law, $(B_t+t)/t\to1$ almost surely, so the same measurable tail event separates the laws.
For completeness the Brownian limit follows from the <Gaussian> bound at integers and the maximal bound on each interval $[n,n+1]$: for every fixed $\eta>0$, both $\sum_n\mathbb P(|B_n|>\eta n)$ and $\sum_n\mathbb P(\sup_{0\le s\le1}|B_{n+s}-B_n|>\eta n)$ are finite. The <Borel-Cantelli lemma> and <continuity> fill the gaps between integers. Intersect over positive rational $\eta$. This proves the limit and the claimed singularity, which is <infinite-horizon singularity of Brownian motion with constant drift>.
On every finite horizon there nevertheless is the density $e^{B_T-T/2}$. Its failure to give a global density is exactly the distinction required by the printed half-line path space.
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