= Solution
For $t\ll D_r^{-1},|\omega|^{-1}$, expand the exact expression:
$$
\boxed{M(t)=4Dt+v^2t^2-\frac{v^2D_r}{3}t^3+O(t^4).}
$$
At asymptotically short time, translational <diffusion> dominates if $D>0$. A ballistic window occurs when $4D/v^2\ll t\ll\min(D_r^{-1},|\omega|^{-1})$, provided these scales are separated. The swimmer then retains nearly constant direction.
For $D_r>0$ and $t\gg D_r^{-1}$, the oscillatory exponential decays and
$$
\boxed{M(t)=4D_{\rm eff}t+\frac{2v^2(\omega^2-D_r^2)}{(D_r^2+\omega^2)^2}+o(1),
\qquad D_{\rm eff}=D+\frac{v^2D_r}{2(D_r^2+\omega^2)}.}
$$
At intermediate times, the angular drift bends persistent trajectories into arcs. If $|\omega|\gg D_r$, several rotations occur before orientational decorrelation, producing oscillatory shoulders rather than a long straight ballistic regime. In the limit $D_r=0$, $\omega\ne0$, the active contribution is bounded and periodic: $2v^2(1-\cos\omega t)/\omega^2$. At $D_r=\omega=0$, it remains ballistic forever, so ordinary late-time active diffusion does not follow.
For $D_r>0$, the central-limit coarse graining of the position gives the late-time <Fokker-Planck equation>
$$
\boxed{\partial_tP(\mathbf r,t)=D_{\rm eff}\nabla^2P(\mathbf r,t).}
$$
There is no surviving drift after orientation averaging. This is an effective large-time, large-distance equation, not the exact closed equation for position at all times.
\Image[/past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2016-chiral-swimmer-displacement.png]
{title=Exact displacement of a persistent chiral swimmer: ballistic and diffusive windows, rotation-induced shoulders, and suppression of active diffusion by angular drift}
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