  Given a K\"ahler manifold $(Z,J,\omega)$ and a compact real submanifold
$M\subset Z$, we study the properties of the gradient map associated with the
action of a noncompact real reductive Lie group ${\rm G}$ on the space of
probability measures on $M.$ In particular, we prove convexity results for such
map when ${\rm G}$ is Abelian and we investigate how to extend them to the
non-Abelian case.
