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For planar Brownian motion started at $(x,y)$ with $y>0$, the probability that its first hit of the real axis lies in $(0,\infty )$ isThis is the bounded harmonic function in the upper half-plane with boundary values zero on the negative half-axis and one on the positive half-axis.

Wiener measure is the probability distribution of Brownian motion on a continuous-path space, usually $C_0([0,T])$ or $C_0([0,\infty ))$.

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 ${\bigcup }_{s\le t}\mathcal{B}(B_s+f(s),r)$. 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.