Divisorial Zariski Decomposition and some properties of full mass currents

Let $alpha$ be a big class on a compact Kahler manifold. We prove that a decomposition $alpha=alpha_1+alpha_2$ into the sum of a modified nef class $alpha_1$ and a pseudoeffective class $alpha_2$ is the divisorial Zariski decomposition of $alpha$ if and only if $operatorname{vol}(alpha)=operatorname{vol}(alpha_1)$. We deduce from this result some properties of full mass currents.