Three equivalent characterizations of probability measures through independence criteria are given. These characterizations lead to a family of Brascamp--Lieb-type inequalities for relative entropy, determine equilibrium states and sharp rates of convergence for certain linear Boltzmann-type dynamics, and unify an assortment of $L^2$ inequalities in probability.

翻译：给出概率测度通过独立性准则的三种等价刻画。这些刻画导出了一族相对熵的Brascamp-Lieb型不等式，确定了特定线性Boltzmann型动力学的平衡态和最优收敛速率，并统一了概率论中一系列$L^2$不等式。