 Initial-value solution of a damped harmonic trap

For distinct roots $r_{\pm }=(-\gamma \pm \sqrt{{\gamma }^2-4mk})/(2m)$, the deterministic response is $X=(v_0-r_{-}{x}_0)e^{r_{+}t}/(r_{+}-r_{-})+(r_{+}{x}_0-v_0)e^{r_{-}t}/(r_{+}-r_{-})$. It satisfies $X(0)=x_0$ and $\dot{X}(0)=v_0$. With zero-mean future noise and prescribed initial data it is also the conditional ensemble mean, not a full stochastic trajectory.