Navier-stokes Equations for a Stochastic Particle Systems on the Lattice

We introduce a class of stochastic models of particles on the cubic lattice Z d with velocities and study the hydrodynamical limit on the diiusive space-time scale. Assuming special initial conditions corresponding to the incompressible regime, we prove that in dimension d 3 there is a law of large numbers for the empirical density and the rescaled empirical velocity eld. Moreover the limit elds satisfy the corresponding incompressible Navier-Stokes equations, with viscosity matrices characterized by a variational formula, formally equivalent to the Green-Kubo formula.