  We discuss peculiarities of quantum fields in de Sitter space on the example
of the self-interacting massive real scalar, minimally coupled to the gravity
background. Non-conformal quantum field theories in de Sitter space show very
special infrared behavior, which is not shared by quantum fields neither in
flat nor in anti-de-Sitter space: in de Sitter space loops are not suppressed
in comparison with tree level contributions because there are strong infrared
corrections. That is true even for massive fields. Our main concern is the
interrelation between these infrared effects, the invariance of the quantum
field theory under the de Sitter isometry and the (in)stability of de Sitter
invariant states (and of dS space itself) under nonsymmetric perturbations.
