Two-dimensional Bessel transition law ($R_t\stackrel{d}{=}\sqrt{(r+\sqrt{t}{G}_1{)}^2+t{G}_2^2}$)

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 $r^2/t$.