Holographic $\beta$ function in de Sitter space

We investigate infrared logarithms in de Sitter space from holographic perspective. We employ a gravitational Fokker-Planck equation to investigate the time evolution of the de Sitter entropy $S=\pi \mathrm{/}(G_N{H}^2)$, where $H$ is the Hubble parameter and $G_N$ is the Newton’s constant. Under the Gaussian approximation, we obtain the dynamical $\beta$ function of $g=1\mathrm{/}S$. It is asymptotically free toward the future. It also possesses the ultraviolet fixed point indicating that the Universe started the de Sitter expansion at the Planck scale. The pre-inflation era exists to prepare a special initial condition for the inflation era as a quantum remnant.