In the first part of the talk, I will recall the essentials of viscosity solutions of first order Hamilton–Jacobi equations in R^d: motivations, definitions, existence, uniqueness and stability. In particular, we shall see that the main tool to analyse these equations is the so-called comparison principle. I will insist upon the main proof of the comparison principle, which relies on a doubling of variables technique.
In the second part of the talk, I will introduce and motivate some recent problems of Hamilton–Jacobi equations set on the space of probability measures. I will present the analogous (and more recent) results of existence, uniqueness and stability. As for the finite dimensional case, we shall see that these properties rely heavily on a comparison principle for a notion of viscosity solutions of these equations.
