  Thermal fluctuations strongly modify the large length-scale elastic behavior
of crosslinked membranes, giving rise to scale-dependent elastic moduli. While
thermal effects in flat membranes are well understood, many natural and
artificial microstructures are modeled as thin elastic {\it shells}. Shells are
distinguished from flat membranes by their nonzero curvature, which provides a
size-dependent coupling between the in-plane stretching modes and the
out-of-plane undulations. In addition, a shell can support a pressure
difference between its interior and exterior. Little is known about the effect
of thermal fluctuations on the elastic properties of shells. Here, we study the
statistical mechanics of shape fluctuations in a pressurized spherical shell
using perturbation theory and Monte Carlo computer simulations, explicitly
including the effects of curvature and an inward pressure. We predict novel
properties of fluctuating thin shells under point indentations and
pressure-induced deformations. The contribution due to thermal fluctuations
increases with increasing ratio of shell radius to thickness, and dominates the
response when the product of this ratio and the thermal energy becomes large
compared to the bending rigidity of the shell. Thermal effects are enhanced
when a large uniform inward pressure acts on the shell, and diverge as this
pressure approaches the classical buckling transition of the shell. Our results
are relevant for the elasticity and osmotic collapse of microcapsules.
