Can the behaviour of microscopic stochastic systems give insight into analysis of upscaled PDEs (part 3)? 

Abstract

Rigorous proofs of singular limits in evolution equations is a hard problem in general, although some fields, for instance singular limits in ODEs and homogenisation theory, are well developed. Starting from the seminal work of Felix Otto and co-authors, it has become evident that a large class of evolution equations (specifically gradient flows) have a natural variational structure which is well-suited for rigorous asymptotic analysis. Furthermore, over the last decade it has become clear that there are deep connections between these (gradient-flow) variational structures and large-deviations of underlying stochastic particle systems. 

In this part 3, I will explore the utility of large-deviations in dealing with singular limit problems in a certain class of (possibly nonlinear) Fokker-Planck equations. The key approach is to study the (Gamma-)convergence of the large-deviation rate functionals, which in addition to convergence of solutions also provide convergence of fluctuations.