  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.
