We propose a new type of SPDEs, singular or with regularized noises, motivated by a study of the fluctuation of the density field in a microscopic interacting particle system. They include a large scaling parameter N, representing the ratio of macroscopic and microscopic sizes, and another scaling parameter \(K=K(N)\) , which controls the formation of the interface of size \(K^{-1/2}\) in the density field. They are derived heuristically from the particle system, assuming the validity of the so-called “Boltzmann-Gibbs principle", that is, a combination of the local ensemble average due to the local ergodicity and its asymptotic expansion. We study a simple situation where the interface is flat and immobile. Under making a proper stretch to the normal direction to the interface, we observe a Gaussian fluctuation of the interface. We also heuristically derive a nonlinear SPDE which describes the fluctuation of the interface.
