A Hamilton-Jacobi equation in the space of probability measures will be introduced as a continuum limit for deterministically interacting particles evolving according to the Newtonian laws. Such equation has been considered before. The well-posedness theory was open unless each individual particle only live in one space dimension. We give a well-posedness result for general dimensions by introducing different new arguments. A key observation is that the space of probability measures has hidden symmetries. We can profit greatly by viewing such space as an infinite dimensional quotient space with the quotient structure expressed using the Wasserstein metric. The quotient structure observation lead us to consider some fine aspects of the optimal transportation calculus that connect with the probabilistic coupling ideas. By using a geometric tangent cone concept to characterized the tangent and co-tangent space and redefine the PDEs, we develops the well-posedness. The cone can be identified with a subset of Markov transition kernels. Its use is critical when a probability measure charge positive mass on small sets (i.e. the phenomenon of condensation).
The talk is based on my joint work with Luigi Ambrosio.