= Solution
<Brownian motion> arises from many microscopic collisions. A one-dimensional Langevin description is
$$
m\dot v=-\zeta v-\kappa x+\xi(t),\qquad
\langle\xi(t)\rangle=0,\qquad
\langle\xi(t)\xi(t')\rangle=2\zeta k_BT\delta(t-t').
$$
The fluctuation-dissipation relation ties random forcing to drag, ensuring <thermal equilibrium>. For a free particle set $\kappa=0$. With initially thermal <velocities>, its mean-square displacement is
$$
\boxed{\langle[x(t)-x(0)]^2\rangle
=2D[t-\tau_v(1-e^{-t/\tau_v})],\qquad
D=\frac{k_BT}{\zeta},\quad \tau_v=\frac m\zeta.}
$$
It is ballistic, $(k_BT/m)t^2$, at short times and diffusive, $2Dt$, for $t\gg\tau_v$. In three dimensions the corresponding long-time result is $6Dt$. At diffusive times, the position propagator from a specified origin is <Gaussian> with <variance> $2Dt$ per coordinate. A small <sphere> in a viscous liquid supplies a physical example; in the slow unbounded-fluid limit its drag is $6\pi\eta_fa$.
For a harmonic trap, overdamped relaxation has time $\tau_\kappa=\zeta/\kappa$, equilibrium <variance> $k_BT/\kappa$ and stationary <correlation>
$$
\langle x(t)x(0)\rangle=\frac{k_BT}{\kappa}e^{-|t|/\tau_\kappa}.
$$
Consequently <stationary trapped Brownian displacement> obeys
$$
\boxed{\langle[x(t)-x(0)]^2\rangle=\frac{2k_BT}{\kappa}(1-e^{-t/\tau_\kappa}).}
$$
Confinement makes this displacement saturate, unlike free <diffusion>. A trajectory initialized exactly at the centre instead has <variance> $(k_BT/\kappa)(1-e^{-2t/\tau_\kappa})$; confusing that conditional <variance> with the stationary displacement gives an erroneous factor of two. The equilibrium position distribution is proportional to $\exp[-\kappa x^2/(2k_BT)]$. More general finite confinement has a geometry-dependent equilibrium distribution and bounded displacement, not necessarily a <Gaussian> one.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2012/paper-1-tsp2012-brownian.png]
{title=Free and trapped Brownian motion}
{description=Original analytic mean-square-displacement curves compare free overdamped diffusion, a stationary harmonic trap and a fixed-origin harmonic-trap variance. The last two have different initial ensembles.}
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