Hessian metric via transport information geometry

We propose to study the Hessian metric of {a functional on the space of probability measures endowed with the Wasserstein $2$-metric}. We name it transport Hessian metric, which contains and extends the classical Wasserstein-$2$ metric. We formulate several dynamical systems associated with transport Hessian metrics. Several connections between transport Hessian metrics and mathematical physics equations are discovered. E.g., the transport Hessian gradient flow, including Newton's flow, formulates a mean-field kernel Stein variational gradient flow; The transport Hessian Hamiltonian flow of Boltzmann-Shannon entropy forms the Shallow water equation; The transport Hessian gradient flow of Fisher information {leads to} the heat equation. {Several examples and closed-form solutions for transport Hessian distances are presented.