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

Entropy Generation at the Horizon Diffuses Cosmological Constant in 2D de Sitter Space