Poincar\'e-Lelong formula, $J$-analytic subsets and Lelong numbers of currents on almost complex manifolds

In this paper, we first establish a Poincar'e-Lelong type formula in the almost complex setting. Then, after introducing the notion of $J$-analytic subsets, we study the restriction of a closed positive current defined in an almost complex manifold $(M,J)$ on a $J$-analytic subset. Finally, we prove that the Lelong numbers of a plurisubharmonic current defined on an almost complex manifold are independent of the coordinate systems.