  The properties of an active fluid, for example, a bacterial bath or a
collection of microtubules and molecular motors, can be accessed through the
dynamics of passive particle probes. Here, in the perspective of analyzing
experimental situations of confinement in droplets, we consider the kinematics
of a negatively buoyant probe particle in an active fluid, both confined within
a spherical domain. The active bath generates a fluctuating flow that pushes
the particle with a velocity that is modeled as a colored stochastic noise,
characterized by two parameters, the intensity and memory time of the active
flow. When the particle departs a little from the bottom of the spherical
domain, the configuration is well approximated by a particle in a
two-dimensional harmonic trap subjected to the colored noise, in which case an
analytical solution exists, which is the base for quantitative analysis. We
numerically simulate the dynamics of the particle and use the planar,
two-dimensional mean square displacement to recover the activity parameters of
the bath. This approach yields satisfactory results as long as the particle
remains relatively confined, that is, as long as the intensity of the colored
noise remains low.
