  Vesicle dynamics in unbounded Poiseuille flow is analyzed using a
small-deformation theory. Our analytical results quantitatively describe
vesicle migration and provide new physical insights. At low ratio between the
inner and outer viscosity $\lambda$ (i.e. in the tank-treading regime), the
vesicle always migrates towards the flow centerline, unlike other soft
particles such as drops. Above a critical $\lambda$, vesicle tumbles and
cross-stream migration vanishes. A novel feature is predicted, namely the
coexistence of two types of nonequilibrium configurations at the centreline, a
bullet-like and a parachute-like shapes.
