Stationary trapped Brownian displacement ($⟨[x(t)-x(0){]}^2⟩=2k_{B}T(1-e^{-t/\tau })/\kappa$)

For overdamped Brownian motion in potential $\kappa x^2/2$, the relaxation time is $\tau =\zeta /\kappa$ and the equilibrium correlation is $(k_{B}T/\kappa )e^{-|t|/\tau }$. Expanding the squared displacement between two equilibrium samples gives the title expression. Initializing the particle exactly at the centre instead gives variance $(k_{B}T/\kappa )(1-e^{-2t/\tau })$. Both are valid but concern different initial ensembles; their saturation values differ by a factor of two. These formulas neglect inertia and require times long compared with momentum relaxation.