In this paper we study the homogenization of a nonautonomous parabolic equation with a large random rapidly oscillating potential in the case of one-dimensional spatial variable. We show that if the potential is a statistically homogeneous rapidly oscillating function of both temporal and spatial variables, then, under proper mixing assumptions, the limit equation is deterministic, and convergence in probability holds. To the contrary, for the potential having a microstructure only in one of these variables, the limit problem is stochastic, and we only have convergence in law.
"Homogenization of a singular random one-dimensional PDE with time-varying coefficients." Ann. Probab. 40 (3) 1316 - 1356, May 2012. https://doi.org/10.1214/11-AOP650