Given the initial state (assumed not to be at the origin) of a Brownian motion in $\mathbb{R}^2$, and the radial component of the process, one can deduce the entire trajectory of the process. See Rogers and Williams' book "Diffusions, Markov Processes and Martingales" for a proof.

Given the initial state (assumed not to be at the origin) of a Brownian motion in $\mathbb{R}^2$, and the angular component of the process in polar coordinates, one can deduce the entire trajectory of the process. See Rogers and Williams' book "Diffusions, Markov Processes and Martingales" for a proof.