  We consider the perturbative treatment of the minimally coupled, massless,
self-interacting scalar field in Euclidean de Sitter space. Generalizing work
of Rajaraman, we obtain the dynamical mass m^2 \propto sqrt{lambda} H^2 of the
scalar for non-vanishing Lagrangian masses and the first perturbative quantum
correction in the massless case. We develop the rules of a systematic
perturbative expansion, which treats the zero-mode non-perturbatively, and goes
in powers of sqrt{lambda}. The infrared divergences are self-regulated by the
zero-mode dynamics. Thus, in Euclidean de Sitter space the interacting,
massless scalar field is just as well-defined as the massive field. We then
show that the dynamical mass can be recovered from the diagrammatic expansion
of the self-energy and a consistent solution of the Schwinger-Dyson equation,
but requires the summation of a divergent series of loop diagrams of
arbitrarily high order. Finally, we note that the value of the long-wavelength
mode two-point function in Euclidean de Sitter space agrees at leading order
with the stochastic treatment in Lorentzian de Sitter space, in any number of
dimensions.
