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 $\mathrm{vol}(\alpha )=\mathrm{vol}({\alpha }_1)$. We deduce from this result some properties of full mass currents.