= Survival among independently moving Poisson traps
{title2=$\mathbb P(T>t)=e^{-\mathbb E\operatorname{vol}(W(t))}$}
Start traps at a unit-intensity <Poisson random measure> in Euclidean space and attach mutually <independent> <Brownian motions>, <independent> of the initial measure. For a deterministic target path $f$, independently thin initial positions according to whether their marked path comes within radius one of the target by time $t$. The retention <probability> $p_t(x)$ integrates, by the <Tonelli theorem>, to the expected volume of the sausage along $f-B$. Symmetry of <Brownian motion> makes this the expected volume of the sausage along $f+B$. The Poisson zero-count <probability> proves the displayed survival formula. This derives the formula directly from <independent> thinning, without assuming a separate marking or displacement theorem.
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