  We obtain a formula for the $n$-dimensional distributions of the Airy$_1$
process in terms of a Fredholm determinant on $L^2(\rr)$, as opposed to the
standard formula which involves extended kernels, on
$L^2(\{1,...,n\}\times\rr)$. The formula is analogous to an earlier formula of
[PS02] for the Airy$_2$ process. Using this formula we are able to prove that
the Airy$_1$ process is H\"older continuous with exponent $\frac12-$ and that
it fluctuates locally like a Brownian motion. We also explain how the same
methods can be used to obtain the analogous results for the Airy$_2$ process.
As a consequence of these two results, we derive a formula for the continuum
statistics of the Airy$_1$ process, analogous to that obtained in [CQR11] for
the Airy$_2$ process.
