  We consider phase transitions on (eternal) de Sitter in an O(N) symmetric
scalar field theory. Making use of Starobinsky's stochastic inflation we prove
that deep infrared scalar modes cannot form a condensate -- and hence they see
an effective potential that allows no phase transition. We show that by proving
convexity of the effective potential that governs deep infrared field
fluctuations both at the origin as well as at arbitrary values of the field.
Next, we present numerical plots of the scalar field probability distribution
function (PDF) and the corresponding effective potential for several values of
the coupling constant at the asymptotic future timelike infinity of de Sitter.
For small field values the effective potential has an approximately quadratic
form, corresponding to a positive mass term, such that the corresponding PDF is
approximately Gaussian. However, the curvature of the effective potential shows
qualitatively different (typically much softer) behavior on the coupling
constant than that implied by the Starobinsky-Yokoyama procedure. For large
field values, the effective potential as expected reduces to the tree level
potential plus a positive correction that only weakly (logarithmically) depends
on the background field. Finally, we calculate the backreaction of fluctuations
on the background geometry and show that it is positive.
