Riemann-Hilbert approach to multi-time processes; the Airy and the Pearcey case

Abstract: We prove that matrix Fredholm determinants related to multi-time processes can be expressed in terms of determinants of integrable kernels `a la Its-Izergin-Korepin-Slavnov (IIKS) and hence related to suitable Riemann-Hilbert problems, thus extending the known results for the single-time case. We focus on the Airy and Pearcey processes. As an example of applications we re-deduce a third order PDE, found by Adler and van Moerbeke, for the two-time Airy process.