I will discuss a new class of PDEs, Hamilton-Jacobi equation in the space of probability measures, in this talk, and report some progress made by my collaborators and myself.

Two types of applications motivate the issue: one is from the probabilistic large deviation study of weakly interacting particle systems in statistical mechanics, another is from an infinite particle version of the variational formulation of Newtonian mechanics.

In creating respective well-posedness theories, two mathematical observations played important roles: One, the free-particle flow picture naturally leads to the use of the optimal mass transportation calculus. Two, there is a hidden symmetry (particle permutation invariance). The space of probability measures in this context is best viewed as an infinite dimensional quotient space. Using a natural metric that has this quotient structure, we are lead to some fine aspects of the transportation calculus that connect with probabilistic coupling ideas. In general, the space of directions for a curve can be a cone instead of a linear space. Such (geometric tangent) cone will be identified with a subset of Markov transition kernels. The use of such cone is critical when a probability measure charge positive mass on small sets (i.e. the phenomenon of condensation).

Time permitting, I will discuss an open issue coming up from the study of the Gibbs-Non-Gibbs transitioning by the Dutch probability community.

The talk is based on my past works with the following collaborators: Markos Katsoulakis, Tom Kurtz, Truyen Nguyen, Andrzej Swiech and Luigi Ambrosio.