 Solution

Interpret the given noise correlation as the three-dimensional dot-product correlation. Each Cartesian component then has $⟨{\xi }_i(t){\xi }_j(t^{\prime })⟩=2\gamma k_{B}T{\delta }_{ij}\delta (t-t^{\prime })$. For the usual Gaussian thermal-noise idealization, each coordinate is an Ornstein-Uhlenbeck process. With $\tau =\gamma /k$,For a particle initially at ${\mathbf{x}}_0=0$,Hence its long-time displacement about the trap/elastic equilibrium hasThis also follows from equipartition theorem: each of three quadratic coordinates contributes $k_{B}T/2$.

A different common experimental quantity is the stationary time-lag displacement. It obeyswith long-lag RMS $\sqrt{6k_{B}T/k}$. The factor of two distinguishes fluctuations about a mean position from differences between two decorrelated positions. The observation convention must be stated rather than silently interchanged.