A Convergence Result for a Planning Problem for Mean Field Games and Rigorous Proof of a Ventcel–Freidlin-Type Large Deviations Principle for the $$1+1$$ KPZ Equation

We study an innovative connection between a very singular optimal transport variational problem with a congestion term connecting two Dirac measures, which is related to the planning problem in mean field games or gases with negative pressure, and a large deviation principle for the KPZ equation. The main result proves a limit of a singular approximate problem with vanishing diffusive flux and a relaxed initial condition. The paper also provides a rigorous justification for the informal analysis of the ad hoc large deviation principle for the KPZ equation appearing in the physics literature.