Divisorial Zariski Decomposition and some properties of full mass currents

Let $\alpha$ be a big class on a compact K\"ahler 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.