On the singularity type of full mass currents in big cohomology classes

Let $X$ be a compact K\"ahler manifold and $\{\theta\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $\theta$-plurisubharmonic functions with full mass are the same as those of the current with minimal singularities. Second, given another big and nef class $\{\eta\}$, we show the inclusion $\mathcal{E}(X,\eta) \cap {PSH}(X,\theta) \subset \mathcal{E}(X,\theta).$ Third, we characterize big classes whose full mass currents are "additive". Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.