Nonintersecting Brownian bridges between reflecting or absorbing walls

We study a model of nonintersecting Brownian bridges on an interval with either absorbing or reflecting walls at the boundaries, focusing on the point in space-time at which the particles meet the wall. These processes are determinantal, and in different scaling limits when the particles approach the reflecting (resp. absorbing) walls we obtain hard-edge limiting kernels which are the even (resp. odd) parts of the Pearcey and tacnode kernels. We also show that in the single time case, our hard-edge tacnode kernels are equivalent to the ones studied by Delvaux [16], defined in terms of a $4\times 4$ Lax pair for the inhomogeneous Painlev\'{e} II equation (PII). As a technical ingredient in the proof, we construct a Schlesinger transform for the $4 \times 4$ Lax pair in [16] which preserves the Hastings--McLeod solutions to PII.