Stochastic approach to chaotic inflation and the distribution of universes

Using the Fokker-Planck equation derived from the stochastic approach to inflation, we investigate the dynamics and global structure of the chaotic inflationary universe. We take full account of the difference in the physical volume of each horizon-size region in the Fokker-Planck equation. We find that the modified Fokker-Planck equation admits a normalizable stationary solution, contrary to the original equation. We evaluate the approximate form of the solution and argue that it describes both the distribution of quantum universes out of which our universe was born and that at large classical (grown-up) universes like ours. In particular, for λφ{symbol}⁴-theory, the distribution of classical universes has a power-law spectrum. Thus there appears no characteristic scale and the distribution of classical universes has a fractal structure.