Fokker-Planck Equation and de Sitter Duality

Abstract: Infra-Red scaling property of inflationary universe is in the same universality class of random walk. The two point correlators of the curvature perturbations are enhanced by the e-folding number N. The distribution function of the curvature perturbation $\rho_t (\zeta)$ satisfies the Fokker Planck equation. The de Sitter universes are dual to the random walk: They belong to the Universality class of dimension two fractal. These boundary and bulk duality are at the heart of holography of quantum gravity. Historically the correspondence of thermodynamics and Einstein's equation are recognized as the first evidence for de Sitter duality .Our de Sitter duality relates the stochastic and geometric point of view. We study two types of the solutions of FP equation in quasi de Sitter space: (1) UV complete spacetime and (2) inflationary spacetime with concave potentials. The maximum entropy principle favors the following scenario: The universe is (a) born with small epsilon and (b) grows by inflation in the concave potential. We predict n_s <0.975(0.97) and r < 0.04(0.03) for N=50(60) at the pivot angle 0.002Mpc^{-1}. We have lowered the upper bound of $r$ by taking account of random walk effect at the boundary. Our predictions are highly consistent with recent observations.