  A stochastic linear transport equation with multiplicative noise is
considered and the question of no-blow-up is investigated. The drift is assumed
only integrable to a certain power. Opposite to the deterministic case where
smooth initial conditions may develop discontinuities, we prove that a certain
Sobolev degree of regularity is maintained, which implies H\"older continuity
of solutions. The proof is based on a careful analysis of the associated
stochastic flow of characteristics.
