= Brownian hitting of lattice spheres
{c}
For three-dimensional <Brownian motion>, every radius $r>0$ and every initial point satisfy
$$
\mathbb P_x\bigl(\exists t\geq0,\ z\in\mathbb Z^3:\ |B_t+z|=r\bigr)=1.
$$
If the initial point is on one of the <spheres>, it is already a hit at time zero. If it is inside a lattice-centered <open ball>, its eventual exit hits that <sphere>. Otherwise choose at each integer time a nearest lattice center. The current displacement from it has norm at most $\sqrt3/2$, so the <multivariate normal distribution> of the next unit increment gives a uniform positive chance of entering its radius-$r/2$ <open ball>. The <geometric tail bound from a uniform escape probability> makes eventual entry certain, and continuity forces a boundary crossing. This periodic-family recurrence does not contradict the <Brownian sphere-hitting probability in dimension three> for one fixed <sphere>.
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