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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 talk, I will first briefly reflect on this seminal relation between (gradient-flow) variational structures and large deviations. I will then focus on using these structures to study asymptotic limits in certain (non-gradient-flow) PDEs. Specifically, I will present these connections for the underdamped Langevin Fokker-Planck equation and focus on the (singular) overdamped limit of this degenerate PDE.