"There is a broad spectrum of problems in scientific disciplines that are dominated by the nonlinear interaction of physical processes across many length and time scales, ranging from the microscopic to the macroscopic. The evolution laws governing such phenomena on macroscopic scales describe quantities which are derived by averaging over many degrees of freedom of a finer length scale. The deviations from this average due to thermal effects or material impurities are often negligible. In some situations, however, they can trigger effects which are observable on the macroscopic length scale. The topic of this proposal is to develop a mathematical methodology which is suitable for a rigorous mathematical analysis of the effects of thermal fluctuations (""noise"") and/or spatial heterogeneities on large spatial and temporal scales by focusing on a special type of partial differential equations, the so-called reaction-diffusion equations. These are widely accepted as modeling important qualitative features of e.g. materials with different phases, but are as well used in other applied areas, like mathematical biology."