Fluctuating Hydrodynamics of the Ising-Kac-Kawasaki Model and Nonlinear Fluctuations Near Criticality

We study the scaling limit behavior of a family of conservative SPDEs as the fluctuating Ising-Kac-Kawasaki dynamics. Precisely, we show that there exists a sequence of the one-dimensional rescaled fluctuating Ising-Kac-Kawasaki equation converges to the solution of the stochastic Cahn-Hilliard equation. This solves a simple version of the conjecture concerning the nonlinear fluctuation phenomenon, proposed by [Giacomin, Lebowitz, Presutti; Math. Surveys Monogr., 1999]. Furthermore, we prove a multi-scale dynamical large deviations in a small noise regime. Finally, we show the $\mathrm{\Gamma }$-convergence of the rate function for the rescaled fluctuating Ising-Kac-Kawasaki equation to the rate function of the Cahn-Hilliard equation.