Bessel Kernel Determinants and Integrable Equations

We derive differential equations for multiplicative statistics of the Bessel determinantal point process depending on two parameters. In particular, we prove that such statistics are solutions to an integrable nonlinear partial differential equation describing isospectral deformations of a Sturm–Liouville equation. We also derive identities relating solutions to the integrable partial differential equation and to the Sturm–Liouville equation which imply an analogue for Painlevé V of Amir–Corwin–Quastel “integro-differential Painlevé II equation”. This equation reduces, in a degenerate limit, to the system of coupled Painlevé V equations derived by Charlier and Doeraene for the generating function of the Bessel process and to the Painlevé V equation derived by Tracy and Widom for the gap probability of the Bessel process. Finally, we study an initial value problem for the integrable partial differential equation. The approach is based on Its–Izergin–Korepin–Slavnov theory of integrable operators and their associated Riemann–Hilbert problems.