  We consider an elliptic and time-inhomogeneous diffusion process with
time-periodic coefficients evolving in a bounded domain of $\mathbb{R}^d$ with
a smooth boundary. The process is killed when it hits the boundary of the
domain (hard killing) or after an exponential time (soft killing) associated
with some bounded rate function. The branching particle interpretation of the
non absorbed diffusion again behaves as a set of interacting particles evolving
in an absorbing medium. Between absorption times, the particles evolve
independently one from each other according to the diffusion semigroup; when a
particle is absorbed, another selected particle splits into two offsprings.
This article is concerned with the stability properties of these non absorbed
processes. Under some classical ellipticity properties on the diffusion process
and some mild regularity properties of the hard obstacle boundaries, we prove
an uniform exponential strong mixing property of the process conditioned to not
be killed. We also provide uniform estimates w.r.t. the time horizon for the
interacting particle interpretation of these non-absorbed processes, yielding
what seems to be the first result of this type for this class of diffusion
processes evolving in soft and hard obstacles, both in homogeneous and
non-homogeneous time settings.
