In recent years the subsolution approach, suggested by P. Dupuis and H. Wang, has proven rather successful in designing efficient algorithms in rare-event simulation. The idea is to relate the performance of the algorithm to a Hamilton-Jacobi equation associated with the large deviations of the underlying stochastic system. Algorithms are then designed from appropriate subsolutions of the Hamilton-Jacobi equation. 

In this talk a duality relation between the Mane's potential and Mather's action functional is derived in the context of convex and state-dependent Hamiltonians. The duality relation is used to obtain new min-max representations of viscosity solutions of first order Hamilton-Jacobi equations.  These min-max representations naturally suggest classes of subsolutions that are good candidates for designing efficient rare-event simulation algorithms. I will show some applications to financial risk management and reliability of power systems. ​