 Integrated Ornstein-Uhlenbeck displacement

For a thermal inertial Langevin equation conditioned on fixed initial position and velocity, the centered displacement is $R_i={\zeta }^{-1}{\int }_0^t[1-e^{-\zeta (t-s)}]A_i(s)ds$. With the fluctuation-dissipation relation for a Langevin particle, $D=k_{B}T/(m\zeta )$ and integrating the squared kernel gives the displayed variance. At short times it is $2D{\zeta }^2{t}^3/3$ per component; at long times it is $2Dt+O(1)$. Averaging initial velocity over thermal equilibrium adds a ballistic contribution and gives $2D[t-(1-e^{-\zeta t})/\zeta ]$ per component.