= Wiener sausage
{c}
{title2=$W_r(t)=\bigcup_{s\leq t}\mathcal B(B_s,r)$}
A Wiener sausage is the region swept out by a ball of fixed radius along a <Brownian motion> path. A deterministic continuous displacement $f(s)$ gives the variant $\bigcup_{s\leq t}\mathcal B(B_s+f(s),r)$. On each bounded time interval it has finite expected volume: the region lies inside a ball whose radius is the path's maximum displacement plus $r$, and the Brownian maximum has Gaussian tail bounds and finite moments of every order.
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