In 2001, Frank Knight constructed a stochastic process modeling the one-dimensional interaction of two particles, one being Newtonian in the sense that it obeys Newton’s laws of motion, and the other particle being Brownian. We construct a multi-particle analog, using Skorohod map estimates in proving a propagation of chaos, and characterizing the hydrodynamic limit as the solution to a PDE with free boundary condition. This PDE resembles the Stefan problem but has a Neumann type boundary condition. Stochastic methods are used to show existence and uniqueness for this free boundary problem.
"Hydrodynamic limit and propagation of chaos for Brownian particles reflecting from a Newtonian barrier." Ann. Appl. Probab. 30 (4) 1582 - 1613, August 2020. https://doi.org/10.1214/19-AAP1536