In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with non-dissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov–Fokker–Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom.
|Number of pages||65|
|Journal||Calculus of Variations and Partial Differential Equations|
|Early online date||28 Jun 2017|
|Publication status||Published - Aug 2017|