Hamilton-Jacobi equations are prevalent throughout mathematics and the study of viscosity solutions to such equations has been an active research area for a number of years, particularly in the weak KAM and dynamical systems communities. Somewhat unexpectedly, in the mid-2000s, Dupuis and Wang identified that certain subsolutions to Hamilton-Jacobi equations in fact play a central role in the design of efficient methods for stochastic simulation. Their approach, referred to as the subsolution approach, however requires the explicit construction of such subsolutions, which in itself can be a difficult problem. In this talk we will discuss how the problem of designing efficient importance sampling algorithms motivated research into a duality between the Mane potential and Mather’s action functional and how this duality leads to a new type of representation of viscosity solutions of rather general Hamilton-Jacobi equations. Moreover, we will see how such representations can indeed lead to efficient simulation algorithms. Time permitted we will also touch on Hamilton-Jacobi equations and viscosity solutions on Wasserstein space.