Real differential forms and currents on p-adic analytic spaces

I will present a joint work with Antoine Chambert-Loir, in which we develop kind of a "harmonic analysis" formalism on Berkovich spaces. More precisely, we define: - real differential forms of bidegree (p,q) on a Berkovich space X of dimension n; - the integral of a (n,n) form (with compact support) on X; - the boundary integral of a (n,n-1) form. We have Stokes and Green formulas in this context. We define currents by duality, and the Poincaré-Lelong formula holds.
We are also able to associate to a metrized line bundle (L,||.||) a curvature form c_1(L,||.||) (if ||.|| is not smooth, this is not a form in general, but a current). If (L,||.||) comes from a formal model, c_1(L,||.||)^n is shown to be a measure, which coincides with a measure previously defined by Chambert-Loir in terms of intersection theory on the special fiber (in his work on p-adic equidistribution of points of small height).