  We study massive real scalar $\phi^4$ theory in the expanding Poincare patch
of de Sitter space. We calculate the leading two-loop infrared contribution to
the two-point function in this theory. We do that for the massive fields both
from the principal and complementary series. As can be expected at this order
light fields from the complementary series show stronger infrared effects than
the heavy fields from the principal one. For the principal series, unlike the
complementary one, we can derive the kinetic equation from the system of
Dyson--Schwinger equation, which allows us to sum up the leading infrared
contributions from all loops. We find two peculiar solutions of the kinetic
equation. One of them describes the stationary Gibbons--Hawking-type
distribution for the density per comoving volume. Another solution shows
explosive (square root of the pole in finite proper time) growth of the
particle number density per comoving volume. That signals the possibility of
the destruction of the expanding Poincare patch even by the very massive
fields. We conclude with the consideration of the infrared divergences in
global de Sitter space and in its contracting Poincare patch.
