  The dynamics and the steady states of a point-like tracer particle immersed
in a confined critical fluid are studied. The fluid is modeled
field-theoretically in terms of an order parameter (concentration or density
field) obeying dissipative or conservative equilibrium dynamics and
(non-)symmetry-breaking boundary conditions. The tracer, which represents,
e.g., a colloidal particle, interacts with the fluid by locally modifying its
chemical potential or its correlations. The coupling between tracer and fluid
gives rise to a nonlinear and non-Markovian tracer dynamics, which is
investigated here analytically and via numerical simulations for a
one-dimensional system. From the coupled Langevin equations for the
tracer-fluid system we derive an effective Fokker-Planck equation for the
tracer by means of adiabatic elimination as well as perturbation theory within
a weak-coupling approximation. The effective tracer dynamics is found to be
governed by a fluctuation-induced (Casimir) potential, a spatially dependent
mobility, and a spatially dependent (multiplicative) noise, the characteristics
of which depend on the interaction and the boundary conditions. The
steady-state distribution of the tracer is typically inhomogeneous. Notably,
when detailed balance is broken, the driving of the temporally correlated noise
can induce an effective attraction of the tracer towards a boundary.
