  We introduce the heat equation relative to a positive dd-bar-closed current
and apply it to the invariant currents associated with Riemann surface
laminations possibly with singularities. The main examples are holomorphic
foliations by Riemann surfaces in projective spaces. We prove two kinds of
ergodic theorems for such currents: one associated to the heat diffusion and
one close to Birkhoff's averaging on orbits of a dynamical system. The heat
diffusion theorem with respect to a harmonic measure is also developed for real
laminations.
