We derive a type of KPZ equation as a scaling limit of fluctuation fields in weakly asymmetric particle systems such as simple exclusion and zero-range processes.  Joint work (in progress) with Milton Jara and Patricia Goncalves.

The need to take stochastic effects into account for modeling complex systems has now become
widely recognized. Stochastic partial differential equations arise naturally as mathematical
models for multiscale systems under random influences. We consider macroscopic dynamics of
microscopic systems described by stochastic partial differential equations. The microscopic
systems are characterized by small scale heterogeneities (spatial domain with small holes or
oscillating coefficients), or fast scale boundary impact (random dynamic boundary condition),
among others.

Effective macroscopic model for such stochastic microscopic systems are derived. The effective
model s are still stochastic partial differential equations, but defined on a unified spatial domain
and the random impact is represented by extra components in the effective models. The
solutions of the microscopic models are shown to converge to those of the effective macroscopic
models in probability distribution, as the size of holes or the scale separation parameter
diminishes to zero. Moreover, the long time effectivity of the macroscopic system in the sense of
convergence in probability distribution, and in the sense of convergence in energy are also
proved.