  We investigate scalar field theories in de Sitter space by means of
nonperturbative renormalization group techniques. We compute the functional
flow equation for the effective potential of O(N) theories in the local
potential approximation and we study the onset of curvature-induced effects as
quantum fluctuations are progressively integrated out from subhorizon to
superhorizon scales. This results in a dimensional reduction of the original
action to an effective zero-dimensional Euclidean theory. We show that the
latter is equivalent both to the late-time equilibrium state of the stochastic
approach of Starobinsky and Yokoyama and to the effective theory for the zero
mode on Euclidean de Sitter space. We investigate the immediate consequences of
this dimensional reduction: symmetry restoration and dynamical mass generation.
