  We introduce fluctuating hydrodynamics approaches on surfaces for capturing
the drift-diffusion dynamics of particles and microstructures immersed within
curved fluid interfaces of spherical shape. We take into account the
interfacial hydrodynamic coupling, traction coupling with the surrounding bulk
fluid, and thermal fluctuations. For fluid-structure interactions, we introduce
Immersed Boundary Methods (IBM) and related Stochastic Eulerian-Lagrangian
Methods (SELM) for curved surfaces. We use these approaches to investigate the
statistics of surface fluctuating hydrodynamics and microstructures. For
velocity autocorrelations, we find characteristic power-law scalings
$\tau^{-1}$, $\tau^{-2}$, and plateaus can emerge. This depends on the physical
regime associated with the geometry, surface viscosity, and bulk viscosity.
This differs from the characteristic $\tau^{-3/2}$ scaling for bulk three
dimensional fluids. We develop theory explaining these observed power-laws
associated with time-scales for dissipation within the fluid interface and
coupling to the surrounding fluid. We then use our introduced methods to
investigate a few example systems and roles of hydrodynamic coupling and
thermal fluctuations including for the kinetics of passive particles and active
microswimmers in curved fluid interfaces.
