  The Airy$_{\beta }$ random point fields ($ \beta = 1,2,4$) are random point
fields emerging as the soft-edge scaling limits of eigenvalues of Gaussian
random matrices. We construct the unlabeled diffusion reversible with respect
to the Airy$_{\beta }$ random point field for each $ \beta = 1,2,4$. We
identify the infinite-dimensional stochastic differential equations (ISDEs)
describing the labeled stochastic dynamics for the unlabeled diffusion
mentioned above. We prove the existence and pathwise uniqueness of strong
solutions of these ISDEs. Furthermore, the solution of the ISDE is the limit of
the solutions of the stochastic differential equations describing the dynamics
of the $ N $-particle system in the soft-edge limit. We thus establish the
construction of the stochastic dynamics whose unlabeled dynamics are reversible
with respect to the Airy random point fields.
  When $ \beta=2 $, the solution equals the stochastic dynamics defined by the
space-time correlation functions obtained by Pr\"ahofer--Spohn, Johansson,
Katori--Tanemura, and Corwin--Hammond, among others. We develop a new method
whereby these ISDEs have unique, strong solutions. We expect that our approach
is valid for other soft-edge scaling limits of stochastic dynamics arising from
the random matrix theory.
