Local behavior and hitting probabilities of the Airy1 process

We obtain a formula for the $n$-dimensional distributions of the Airy${}_1$ process in terms of a Fredholm determinant on $L^2(\text{rr})$, as opposed to the standard formula which involves extended kernels, on $L^2(\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}\times \text{rr})$. 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 $\frac{1}{2}-$ 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.