Well-posedness of Hamilton-Jacobi equations with application in Large deviations theory

We prove the comparison principle for viscosity solutions of Hamilton-Jacobi equations with Hamiltonians that have a representation as the supremum of a difference of two functions: an internal Hamiltonian depending on a control variable and a function interpreted as a cost of applying the controls.
Our major innovation lies in the use of a cost function that can be discontinu- ous, unbounded, and depending on momenta, enabling us to address previously unexplored scenarios, such as cases arising from the theory of large deviations and homogenisation.
After a brief introduction of the theory that connects large deviations to Hamilton-Jacobi equations (developed by J. Feng and T.K. Kurtz in 2006), we apply the method to prove large deviations for Switching Markov processes and multi-scale Markov processes.
The talk is based on two joint works with Richard C. Kraaij (TU Delft).