Holographic <i>β</i> Function in de Sitter Space

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