  A system of non-intersecting squared Bessel processes is considered which all
start from one point and they all return to another point. Under the scaling of
the starting and ending points when the macroscopic boundary of the paths
touches the hard edge, a limiting critical process is described in the
neighbourhood of the touching point which we call the hard edge tacnode
process. We derive its correlation kernel in an explicit new form which
involves Airy type functions and operators that act on the direct sum of
$L^2(\mathbb R_+)$ and a finite dimensional space. As the starting points of
the squared Bessel paths are set to 0, a cusp in the boundary appears. The
limiting process is described near the cusp and it is called the hard edge
Pearcey process. We compute its multi-time correlation kernel which extends the
existing formulas for the single-time kernel. Our pre-asymptotic correlation
kernel involves the ratio of two Toeplitz determinants which are rewritten
using a Borodin-Okounkov type formula.
