  The scale invariance of the universe is slightly broken by slow roll
parameters. It is likely the slow roll is dual to the random walk. We
investigate the distribution function of the conformal zeromode. We identify de
Sitter entropy $S_{dS}$ with the distribution entropy of the conformal zeromode
$\rho(\omega)$. We have collected convincing support on our postulate. The
semiclassical evidence is that the both are given by the gravitational coupling
$1/g=\log N/2$ where $g=G_NH^2/\pi$ and $N$ is the e-folding number. We show
the renormalized distribution function obeys gravitational Fokker-Planck
equation (GFP) and Langevin equations . Under the Gaussian approximation, they
boil down to a simple first order partial differential equation. The identical
equation is derived by the thermodynamic arguments in the inflationary
space-time. GFP determines the evolution of de Sitter entropy of the universe.
It coincides with $\beta$ function of $g$. We find two types of the solutions
of GFP:(1) UV complete spacetime and (2) inflationary spacetime with power
potentials. The maximum entropy principle favors the scenario: (a) born small
$\epsilon$ and (b) grow large by inflation. We like to convey the emerging
notion of de Sitter duality. The inflationary universe: (bulk/geometrical) is
dual to the stochastic space-time on the boundary (cosmological horizon ) as
the both are the solutions of GFP.
