= Two-dimensional Bessel transition law
{title2=$R_t\overset d=\sqrt{(r+\sqrt t\,G_1)^2+tG_2^2}$}
A dimension-two <Bessel process> starting at $r>0$ has the displayed marginal law, where $G_1,G_2$ are independent standard <normal random variables>. Take the norm of a <planar Brownian motion> starting at $(r,0)$. <Planar Brownian motion avoids a fixed point>, so the norm stays positive; <Itô formula> gives drift $1/(2R)$ and a martingale term of bracket $t$. The <Lévy characterization of Brownian motion> identifies that term as its driving Brownian motion. The squared radius divided by $t$ has a <noncentral chi-squared distribution> with two degrees of freedom and noncentrality $r^2/t$.
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