Uniqueness and short time regularity of the weak Kähler-Ricci flow

Let X be a compact Kähler manifold. We prove that the Kähler-Ricci flow starting from arbitrary closed positive (1,1)-currents is smooth outside some analytic subset. This regularity result is optimal, meaning that the flow has positive Lelong numbers for short time if the initial current has. We also prove that the flow is unique when starting from currents with zero Lelong numbers.