In previous works we understood the microscopic derivation of a family of Non-Linear Stochastic Equations (NLSE), by analyzing two different kinds of one-dimensional spin models . We proved that the "critical fluctuations" of the "magnetization density field" converge to the solutions of these kinds of NLSE. These NLSE have applications in the theory of critical phenomena (as is evident also from the models) and problems concerning stochastic quantization. The invariant measures of these NLSE are the P( ) Euclidean Fields. The interesting problems concerning this topic are in space -dimension 2.
For one of these two models analyzed in the previous works we might have good possibilities to extend the results in 2-space dimensions. This model evolves with a reversible "Glauber dynamics" with respect to a Gibbs measure defined by a "Kac potential".
Recent results obtained by Cassandro M., Marra R., Presutti E. concerning the invariant measure in 2 dimensions let us think that we should succeed in analyzing the dynamics of the 2-dimensional critical fluctuations. Difficulties arize because of ultra-violet divergencies