  We investigate a solution of the exactly renormalized Liouville action to
foresee the fate of the two-dimensional de Sitter space. We work in the
semiclassical region with a large matter central charge $c$. Instead of de
Sitter expansion, it performs a slow-roll inflation with the parameters
$\epsilon=(1/2)\eta =6/c$. An inflaton field is induced in the effective theory
to describe quantum effects of the Liouville theory. The geometric entropy
increases logarithmically with the Hubble radius. We propose that de Sitter
entropy is carried by superhorizon modes of the metric. It can be directly
estimated from the partition function as $S=\log Z$ in Liouville gravity. We
formulate a gravitational Fokker-Planck equation to elucidate the Brownian
process at the horizon: the superhorizon modes are constantly jolted by
newcomers. We show that such a built-in entropy-generating process diffuses the
cosmological constant. We evaluate von Neumann entropy associated with the
distribution function of superhorizon modes. It always increases under the
Fokker-Planck equation in a consistent way with semiclassical estimates. The
maximum entropy principle operates in quantum gravity. An analogous entropy
production mechanism at the horizon might have increased the Hubble radius much
beyond the microscopic physics scale in the Universe.
