We generalize the force-level, microscopic, Nonlinear Langevin Equation (NLE) theory and its elastically collective generalization (ECNLE theory) of activated dynamics in bulk spherical particle liquids to address the influence of random particle pinning on structural relaxation. The simplest neutral confinement model is analyzed for hard spheres where there is no change of the equilibrium pair structure upon particle pinning. As the pinned fraction grows, cage scale dynamical constraints are intensified in a manner that increases with density. This results in the mobile particles becoming more transiently localized, with increases of the jump distance, cage scale barrier and NLE theory mean hopping time; subtle changes of the dynamic shear modulus are predicted. The results are contrasted with recent simulations. Similarities in relaxation behavior are identified in the dynamic precursor regime, including a roughly exponential, or weakly supra-exponential, growth of the alpha time with pinning fraction and a reduction of dynamic fragility. However, the increase of the alpha time with pinning predicted by the local NLE theory is too small, and severely so at very high volume fractions. The strong deviations are argued to be due to the longer range collective elasticity aspect of the problem which is expected to be modified by random pinning in a complex manner. A qualitative physical scenario is offered for how the three distinct aspects that quantify the elastic barrier may change with pinning. ECNLE theory calculations of the alpha time are then presented based on the simplest effective-medium-like treatment for how random pinning modifies the elastic barrier. The results appear to be consistent with most, but not all, trends seen in recent simulations. Key open problems are discussed with regards to both theory and simulation.
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