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.

翻译：我们提议研究以瓦塞斯坦为单位的概率测量空间的赫森测量值的赫森测量值,用2美元的量度来计算。我们称之为运输赫森测量值,它包含并扩展了古典瓦西斯坦为单位的赫森测量值。我们开发了与运输赫森测量值相关的若干动态系统。发现了赫森测量值与数学物理方程之间的若干连接。例如,包括牛顿流在内的赫森梯度的运输流,形成了一个中位内核的斯坦因变异梯度流;波尔兹曼-尚农的海森汉密尔顿流构成浅水方程的运输;渔业信息的赫森梯度流动导致热方程。提出了赫森距离运输的几个例子和封闭式解决办法。