Elliptic equations for measures on infinite dimensional spaces and applications

We introduce and study a new concept of a weak elliptic equation for measur es on infinite dimensional spaces. This concept allows one to consider equa tions whose coefficients are not globally integrable. By using a suitably e xtended Lyapunov function technique, we derive a priori estimates for the s olutions of such equations and prove new existence results. As an applicati on, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes e quations and investigate the elliptic equations for the corresponding invar iant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gi bbs distributions and prove an existence result applicable to a wide class of models.