  The slow rolling inflation is dual to the random walk of conformal zero-mode.
The 2 dimensional Fokker-Planck theory predicts the slow roll parameters of 4d
inflation theory. The O(N) enhancements of the two point functions, N is the
e-folding number, suppress the slow roll parameters by the same magnitude.
Under the gaussian approximation, FP equation boils down to a solvable first
order partial differential equation. The identical equation is derived by the
thermodynamic arguments . We study two types of the solutions of :(1) UV
complete spacetime and (2) inflationary spacetime with power potentials. The
concavity of entangled entropy dictates the potential of inflation is also
concave. The maximum entropy principle favours the scenario: the universe is
(a) born small and (b) grows large by inflation in the concave potentials. We
predict 1-ns > 0.02(0.016) and r < 0.08(0.066) for N = 50(60)at the pivot angle
0.002(Mpc)-1. we propose a scenario to produce the curvature perturbation in
the right ball park.
