  The scale invariance of the quantum fluctuations in de Sitter space leads to
the appearance of de Sitter symmetry breaking infra-red logarithms in the
graviton propagator. We investigate physical effects of soft gravitons on the
local dynamics of matter fields well inside the cosmological horizon. We show
that the IR logarithms do not spoil Lorentz invariance in scalar and Dirac
field theory. The leading IR logarithms can be absorbed by a time dependent
wave function renormalization factor in the both cases. In the interacting
field theory with $\lambda \phi^4$ and Yukawa interaction, we find that the
couplings become time dependent with definite scaling exponents. We argue that
the relative scaling exponents of the couplings are gauge invariant and
physical as we can use the evolution of a coupling as a physical time.
