  The stability of photon trajectories in models of the Universe that have
constant spatial curvature is determined by the sign of the curvature: they are
exponentially unstable if the curvature is negative and stable if it is
positive or zero. We demonstrate that random fluctuations in the curvature
provide an additional stabilizing mechanism. This mechanism is analogous to the
one responsible for stabilizing the stochastic Kapitsa pendulum. When the mean
curvature is negative it is capable of stabilizing the photon trajectories;
when the mean curvature is zero or positive it determines the characteristic
frequency with which neighbouring trajectories oscillate about each other. In
constant negative curvature models of the Universe that have compact topology,
exponential instability implies chaos (e.g. mixing) in the photon dynamics. We
discuss some consequences of stochastic stabilization in this context.
