  The Airy$_\beta$ point process, originally introduced by Ram\'irez, Rider,
and Vir\'ag, is defined as the spectrum of the stochastic Airy operator
$\mathcal{H}_\beta$ acting on a subspace of $L^2[0,\infty)$ with Dirichlet
boundary condition. In this paper we study the coupled family of point
processes defined as the eigenvalues of $\mathcal{H}_\beta$ acting on a
subspace of $L^2[t,\infty)$. These point processes are coupled through the
Brownian term of $\mathcal{H}_\beta$. We show that these point processes as a
function of $t$ are differentiable with explicitly computable derivative.
Moreover when recentered by $t$ the resulting point process is stationary. This
process can also be viewed as an analogue to the 'GUE minor process' in the
tridiagonal setting.
