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We give a new sufficient condition of the quasi-Gibbs property. This resultis a refinement of one given in a previous paper (\cite{o.rm}), and will beused in a forth coming paper to prove the quasi-Gibbs property of Airy randompoint fields (RPFs) and other RPFs appearing under soft-edge scaling. Thequasi-Gibbs property of RPFs is one of the key ingredients to solve theassociated infinite-dimensional stochastic differential equation (ISDE).Because of the divergence of the free potentials and the interactions of thefinite particle approximation under soft-edge scaling, the result of theprevious paper excludes the Airy RPFs, although Airy RPFs are the mostsignificant RPFs appearing in random matrix theory. We will use the result ofthe present paper to solve the ISDE for which the unlabeled equilibrium stateis the $\mathrm{Airy}_{\beta}$ RPF with $ \beta = 1,2,4 $.

Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field