Bridging scales: from microscopic dynamics to macroscopic laws

One fascinating aspect of probability theory is the universal aspect of the objects it allows us to construct.  The most well-known example of this phenomenon is the central limit theorem: for a very large class of collections of random variables, additive functionals that only depend weakly on any one element of the collection exhibit Gaussian behaviour in the limit.  When taking time evolution into account, it turns out that in certain "cross-over regimes" the large-scale behaviour of a number of stochastic systems can formally be described by an ill-posed stochastic PDE.