= Three-dimensional Bessel process
{title2=$dR_t=dB_t+R_t^{-1}dt$}
The radial part of three-dimensional <Brownian motion> started away from the origin is a <Bessel process> of dimension three. The radial <Laplacian> is $\Delta|x|=2/|x|$, so the <Itô formula> gives the displayed equation. The radial martingale has unit <quadratic variation> rate and is <Brownian motion> by the <Lévy characterization of Brownian motion>. The reciprocal $1/R$ is a <local martingale>. Stopping it on $(\varepsilon,M)$ gives $\mathbb P_x(T_\varepsilon<T_M)=\varepsilon(M-x)/(x(M-\varepsilon))$. Letting $\varepsilon\downarrow0$ shows that zero is not hit.
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