The most likely evolution of interacting particles between two densities is a Markov diffusion solving the hydrodynamic variational problem, with new current-constrained variants included.
Seo; Large-deviation principle for interacting Brownian motions
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Macroscopic fluctuation theory from a Lagrangian viewpoint and the Schr\"odinger problem
The most likely evolution of interacting particles between two densities is a Markov diffusion solving the hydrodynamic variational problem, with new current-constrained variants included.